Home » Mathematics » Pre Calculus » Determinants & Inverses of Matrices

Vincent Selhorst-Jones

Vincent Selhorst-Jones

Determinants & Inverses of Matrices

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Table of Contents

Section 1: Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
Section 2: Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
The Basketball Factory
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
Section 3: Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
Section 11: Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
Section 14: Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
Section 17: Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
Think About the Results
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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Lecture Comments (11)

1 answer

Last reply by: Ann Gao
Tue Jul 28, 2020 2:41 PM

Post by Ann Gao on July 28, 2020

At 14:40, when we were calculating the minor, how did 10-9 become -1 ?

2 answers

Last reply by: Leonardo Luo
Thu May 28, 2020 7:11 AM

Post by Leonardo Luo on May 14, 2020

1/0 is infinity, right?

1 answer

Last reply by: Professor Selhorst-Jones
Thu Dec 4, 2014 5:25 PM

Post by Johnathon Kocher on December 3, 2014

Hello, I'm just dropping a comment to let you know the "Download Lecture Slides" and "Practice Questions" tabs are empty for this lecture.

1 answer

Last reply by: Professor Selhorst-Jones
Sun Mar 9, 2014 2:34 PM

Post by Christopher Hu on March 8, 2014

I tried the "alternative" method for a 4x4 matrix and it does not work. the "alternative" method only works for a 3x3, right?

1 answer

Last reply by: Professor Selhorst-Jones
Sat Aug 10, 2013 1:00 PM

Post by Ikze Cho on August 10, 2013

Does the "alternative method for 3x3 matrices" only apply for the 3x3 matrices or also for other matrices?

Determinants & Inverses of Matrices

  • The inverse of a matrix A is some matrix A−1 such that when we multiply them together we get the identity matrix, I. In other words, they "cancel" each other.
  • Not all matrices can be inverted. A matrix that can be is called invertible (or `nonsingular'). If it can not be inverted, it is singular. To be invertible, a matrix must have these properties:
    • The matrix must be square,
    • The determinant of the matrix must be nonzero.
  • The determinant is a real number associated with a square matrix. The determinant of a matrix A is denoted by either  det(A)  or |A|. [Although |A| may look similar to absolute value, it is the determinant of A and can produce any real number (including negative numbers).] If the determinant of a matrix is nonzero, the matrix is invertible, and vice-versa.

    det
     
    		(A) ≠ 0    
    ⇔
        A is invertible

    det
     
    		(A) = 0    
    ⇔
        A is not invertible
  • The determinant of a 2×2 matrix
    A =⎡
    ⎢
    ⎢
    ⎣
    a
    b
    c
    d
    		⎤
    ⎥
    ⎥
    ⎦



     
    is given by

    det
     
    		(A) = |A| = ad − bc.
    A good mnemonic to remember this is to think in terms of diagonals: the down diagonal multiplied together then subtracted by the up diagonal (also multiplied).
  • To take the determinant of a larger matrix, we need the two following concepts.
    • Minors: For a square matrix A, the minor Mij of the entry aij is the determinant of the matrix obtained by deleting the ith row and jth column.
    • Cofactors: The cofactor is very closely based on the minor. It just multiplies the minor by 1 or −1 based on the location of the entry the minor comes from. The cofactor Cij of the entry aij is given by
      Cij = (−1)i+j Mij.
      We can also see this as an alternating sign pattern. It's very similar to a chessboard where one color is positive and the other is negative. See the video for a visual reference.
    The determinant of an n ×n matrix A is given by the sum of the entries in any row or column each multiplied by their respective cofactor.

    det
     
    		(A)  =  |A|
    =
    ak1 Ck1 + ak2Ck2 + …+ akn Ckn
    =
    a1k C1k + a2kC2k + …+ ank Cnk
    Note that this is true for any value of k (as long as 1 ≤ k ≤ n). That is, we can choose to do this process with any row or column and get the same result.
  • For a 2×2 matrix
    A =⎡
    ⎢
    ⎢
    ⎣
    a
    b
    c
    d
    		⎤
    ⎥
    ⎥
    ⎦



     
    		,
    the inverse of A (if det(A) ≠ 0) is
    A−1 =1
    ad−bc


     
    		⎡
    ⎢
    ⎢
    ⎣
    d
    −b
    −c
    a
    		⎤
    ⎥
    ⎥
    ⎦



     
    		=1

    det
     
    		(A)


     
    		⎡
    ⎢
    ⎢
    ⎣
    d
    −b
    −c
    a
    		⎤
    ⎥
    ⎥
    ⎦



     
    		.
  • For the most part, at the level of this course or any similar math class, you will probably not need to compute the inverse of a matrix any larger than 2 ×2. If for some reason you need to calculate the inverse of a matrix that is larger than 2×2 and you must do it by hand, see the bottom of these notes.
  • Given A and A−1, we have A−1 A B = B: they cancel each other out and have no net effect. This is because
    A−1 A  = I = A A−1.
    When multiplied together, they create the identity matrix I, which (as noted in the previous lesson) has no effect in multiplication.
  • It is important to note that if we multiply an equation by a matrix on both sides, we must choose a direction to multiply from and do the same for both sides of the equation. We must multiply on the left or on the right for both sides. [This is because PQ ≠ QP for most matrices.]
  • To find the inverse of larger matrices by hand, we will need some techniques we haven't learned just yet. In the first part of the next lesson, we discuss augmented matrices, row operations, and Gauss-Jordan elimination. If you're not familiar with these things, go check them out first. Here are the steps to find an inverse for any size matrix:

      1. For an n ×n matrix A, begin by creating an augmented matrix with the identity matrix In:
      ⎡
      ⎣
       
      A
      In
      		⎤
      ⎦
       

      2. Apply the method of Gauss-Jordan elimination (use row operations) to reduce A (the left side) to I. The result of the augmented matrix will be
      ⎡
      ⎣
       
      In
      A−1
      		⎤
      ⎦
       

      3. Finally, check your work. It's very easy to make a mistake in all that arithmetic, so check by showing either
      A−1 A = I       or        AA−1 = I.

Determinants & Inverses of Matrices

Find the determinant of A = [
5
−3
6
2
]
  • The determinant of a matrix is a real number associated with that matrix. (The determinant only exists for square matrices.) For a matrix A, the determinant of the matrix can be written as det(A) or |A|.
  • For a 2×2 matrix, the determinant is is given as
    A =⎡
    ⎢
    ⎢
    ⎣
    a
    b
    c
    d
    		⎤
    ⎥
    ⎥
    ⎦    		    ⇒        det
    		(A) = |A| =     ad−bc
  • Thus, we plug in based on the above formula for the determinant:
    det
    		(A)     =     5 ·2 − (−3) ·6     =     28
det(A) = 28
Find the value of |
7
6
5
2
|.
  • If a matrix has large absolute value bars on either side instead of brackets, that is short-hand for saying it is the determinant of that matrix. Thus, for this problem, we're just looking for the determinant of that matrix.
  • For a 2×2 matrix, the determinant is is given as
    A =⎡
    ⎢
    ⎢
    ⎣
    a
    b
    c
    d
    		⎤
    ⎥
    ⎥
    ⎦    		    ⇒        det
    		(A) = |A| =     ad−bc
  • Thus, we plug in based on the above formula for the determinant:
    ⎢
    ⎢
    ⎢
    ⎢
    7
    6
    5
    2
    		⎢
    ⎢
    ⎢
    ⎢    		    =     7 ·2 − 6 ·5     =     −16
−16
Is it possible to invert the matrix A = [
8
20
18
45
]?
  • For a matrix to be invertible (able to be inverted), it must have a non-zero determinant.
    det
    		(A) ≠ 0   ⇔   A is invertible    ⎢
    ⎢    		    det
    		(A) = 0   ⇔   A is not invertible
    Thus, to check if A can be inverted, we must find its determinant.
  • For a 2×2 matrix, the determinant is is given as
    A =⎡
    ⎢
    ⎢
    ⎣
    a
    b
    c
    d
    		⎤
    ⎥
    ⎥
    ⎦    		    ⇒        det
    		(A) = |A| =     ad−bc
  • Thus, we plug in based on the above formula for the determinant:
    det
    		(A)     =     8 ·45 − 20 ·18     =     360 − 360     =     0
    Therefore, since the determinant is 0, it can not be inverted.
No, it is not possible to invert A.
Is A = [
8
4
3
2
] invertible? If so, what is A−1?
  • First, determine if A is invertible by making sure det(A) ≠ 0. For a 2×2 matrix, the determinant is is given as
    A =⎡
    ⎢
    ⎢
    ⎣
    a
    b
    c
    d
    		⎤
    ⎥
    ⎥
    ⎦    		    ⇒        det
    		(A) = |A| =     ad−bc
    Thus, for our matrix, we have
    det
    		(A)     =     8 ·2 − 4 ·3     =     4.
    Since det(A) = 4 ≠ 0, we see that A can be inverted.
  • For a 2 ×2 matrix, the inverse of the matrix (if det(A) ≠ 0) is as follows:
    A−1     =     1
    det
    		(A)
    		⎡
    ⎢
    ⎢
    ⎣
    d
    −b
    −c
    a
    		⎤
    ⎥
    ⎥
    ⎦    		    =     1
    ad−bc
    		⎡
    ⎢
    ⎢
    ⎣
    d
    −b
    −c
    a
    		⎤
    ⎥
    ⎥
    ⎦
  • Thus, since we already found the determinant of our matrix (det(A) = 4), we can just plug in and find A−1:
    A−1     =     1
    4
    		·⎡
    ⎢
    ⎢
    ⎣
    2
    −4
    −3
    8
    		⎤
    ⎥
    ⎥
    ⎦    		    =    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    −1
    − 3
    4
    2
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
Yes, A can be inverted: A−1 = [
[1/2]
−1
−[3/4]
2
].
Find the inverse of X = [
2
−5
−3
9
], then show your answer works as the inverse.
  • Start off by finding X−1. The problem basically guarantees that it exists, but it's still useful to find the determinant of X since we'll use it later:
    det
    		(X)     =     2 ·9 − (−5)·(−3)     =     3
    Great, so X definitely does have an inverse (since det(X) ≠ 0). Now we work to find the inverse.
  • For a 2 ×2 matrix, the inverse of the matrix (if det(A) ≠ 0) is as follows:
    A−1     =     1
    det
    		(A)
    		⎡
    ⎢
    ⎢
    ⎣
    d
    −b
    −c
    a
    		⎤
    ⎥
    ⎥
    ⎦    		    =     1
    ad−bc
    		⎡
    ⎢
    ⎢
    ⎣
    d
    −b
    −c
    a
    		⎤
    ⎥
    ⎥
    ⎦
    Therefore, we follow the same structure to find X−1:
    X−1     =     1
    3
    		·⎡
    ⎢
    ⎢
    ⎣
    9
    5
    3
    2
    		⎤
    ⎥
    ⎥
    ⎦    		    =    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    3
    5
    3
    1
    2
    3
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
  • To show that our answer of X−1 truly is the inverse of X, we must show that it works. That is to say, X X−1 = I and X−1 X = I: when we multiply X and its inverse, we get the identity matrix (1's on the main diagonal, 0's everywhere else). Show what we get for X and X−1 multiplied together, done from both possible sides (since matrix multiplication gives different results based on which side the matrix multiplies from). [Note: If you're not currently comfortable with doing matrix multiplication, make sure to go watch the previous lesson that introduces matrices for the first time and explains how matrix multiplication works in detail. Understanding matrix multiplication is absolutely crucial to working with matrices.]
    X  X−1              ⎢
    ⎢    		              X−1 X

    ⎡
    ⎢
    ⎢
    ⎣
    2
    −5
    −3
    9
    		⎤
    ⎥
    ⎥
    ⎦    		⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    3
    5
    3
    1
    2
    3
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦    		      ⎢
    ⎢    		      ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    3
    5
    3
    1
    2
    3
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦    		⎡
    ⎢
    ⎢
    ⎣
    2
    −5
    −3
    9
    		⎤
    ⎥
    ⎥
    ⎦

    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    2 ·3−5 ·1
    2· 5
    3
    		− 5 · 2
    3
    −3 ·3 + 9 ·1
    −3· 5
    3
    		 + 9 · 2
    3
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦    		      ⎢
    ⎢    		      ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    3 ·2 + 5
    3
    		·(−3)
    3·(−5) + 5
    3
    		·9
    1 ·2+ 2
    3
    		·(−3)
    1 ·(−5) + 2
    3
    		·9
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦

    ⎡
    ⎢
    ⎢
    ⎣
    1
    0
    0
    1
    		⎤
    ⎥
    ⎥
    ⎦    		          ⎢
    ⎢    		          ⎡
    ⎢
    ⎢
    ⎣
    1
    0
    0
    1
    		⎤
    ⎥
    ⎥
    ⎦
    Great! They both come out to be I, the identity matrix, so we've shown that our inverse of X−1 works exactly like it should.
X−1 = [
3
[5/3]
1
[2/3]
] To show that X−1 works, multiply it against X from the left and right sides to show that the result comes out as I, the identity matrix.
[Technically, it is enough to show that X−1 works on one side of X, either left or right. If it can be shown that X X−1 = I, multiplying from the other side must give the same result too (and vice-versa). That said, some teachers and books might not prove this fact or allow you to use it, requiring you to instead show that both X X−1 and X−1X come out to be I. What is allowed will vary depending on where you take the course, so it is normally better to be careful and do both sides when asked to verify an inverse on a test.]

A =⎡
⎢
⎢
⎣
8
3
10
4
⎤
⎥
⎥
⎦		              B =⎡
⎢
⎢
⎣
−5
3
−1
0
⎤
⎥
⎥
⎦
In the equation  2B = XA, solve for X, and give it as a matrix array.
  • We can approach matrix equations very similarly to how we approach normal equations. For the matrix equation 2B=XA, we're looking to isolate the X on one side of the equation. The issue we must deal with is that X and the A are connected through matrix multiplication. Thus, we need something that can cancel out the A. To cancel out a matrix, we use its inverse. Since A A−1 = I, if we multiply both sides of the equation by A−1, we will cancel out the A on the right side, leaving us with only X. (This works because I [the identity matrix] multiplied on any matrix has no effect. It is similar to multiplying any number by 1.) However, it is crucial to note that when we multiply by A−1 on both sides we must multiply by it on the right side. (This is because if we multiply on the left, it would "hit" the X instead of canceling out the A.) Because we multiply by A−1 on the right side for one half of the equation, we must multiply on the right side for both halves of the equation. If we multiply on different sides, the equation will no longer be equal, since matrix multiplication "cares" about which side it is applied on.
  • With this in mind, we solve the equation as
    2B  = XA

    2B   ·A−1    =   XA   ·A−1

    2BA−1  = X             
    Since we were told to find X as a matrix array (that is, a rectangular array with numbers inside), we now need to calculate the value of 2BA−1.
  • Begin by finding A−1. For a 2 ×2 matrix, the inverse of the matrix (if det(A) ≠ 0) is as follows:
    A−1     =     1
    det
    		(A)
    		⎡
    ⎢
    ⎢
    ⎣
    d
    −b
    −c
    a
    		⎤
    ⎥
    ⎥
    ⎦    		    =     1
    ad−bc
    		⎡
    ⎢
    ⎢
    ⎣
    d
    −b
    −c
    a
    		⎤
    ⎥
    ⎥
    ⎦
    Thus, working with the A for this problem, we find
    A−1     =     1
    8 ·4 − 3 ·10
    		⎡
    ⎢
    ⎢
    ⎣
    4
    −3
    −10
    8
    		⎤
    ⎥
    ⎥
    ⎦    		 ,
    then simplify:
    A−1 =     1
    2
    		⎡
    ⎢
    ⎢
    ⎣
    4
    −3
    −10
    8
    		⎤
    ⎥
    ⎥
    ⎦    		    =    ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    2
    − 3
    2
    −5
    4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
  • We want to know the value of X in
    X = 2BA−1,
    and since we know both B and A−1, we can plug in to find X.
    X =     2⎡
    ⎢
    ⎢
    ⎣
    −5
    3
    −1
    0
    		⎤
    ⎥
    ⎥
    ⎦    		⎡
    ⎢
    ⎢
    ⎢
    ⎣
    2
    − 3
    2
    −5
    4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    Apply the scalar multiple:
    X =    ⎡
    ⎢
    ⎢
    ⎣
    −10
    6
    −2
    0
    		⎤
    ⎥
    ⎥
    ⎦    		⎡
    ⎢
    ⎢
    ⎢
    ⎣
    2
    − 3
    2
    −5
    4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    Then carry out the matrix multiplication. [If you're unfamiliar with matrix multiplication or find it particularly difficult, make sure to watch the previous lesson where it is explained in detail. Knowing how to do matrix multiplication is crucial to success in understanding matrices.]
    X =    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    −10 ·2 + 6 ·(−5)
    −10 ·(− 3
    2
    		 )+6 ·4
    −2 ·2 + 0 ·(−5)
    −2·(− 3
    2
    		 ) + 0 ·4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
    Then simplify to find the answer:
    X  =  ⎡
    ⎢
    ⎢
    ⎣
    −50
    39
    −4
    3
    		⎤
    ⎥
    ⎥
    ⎦
  • Finally, after a problem as difficult and lengthy as this, it's a great idea to check your answer. Matrix work involves a lot of tiny calculations and steps, so it can be easy to make a mistake. Counter this by checking your answer. From the problem, we have
    2B = XA,
    so if we plug everything into this, each side should come out the same.
    2⎡
    ⎢
    ⎢
    ⎣
    −5
    3
    −1
    0
    		⎤
    ⎥
    ⎥
    ⎦    		    =    ⎡
    ⎢
    ⎢
    ⎣
    −50
    39
    −4
    3
    		⎤
    ⎥
    ⎥
    ⎦    		⎡
    ⎢
    ⎢
    ⎣
    8
    3
    10
    4
    		⎤
    ⎥
    ⎥
    ⎦

    ⎡
    ⎢
    ⎢
    ⎣
    −10
    6
    −2
    0
    		⎤
    ⎥
    ⎥
    ⎦    		    =    ⎡
    ⎢
    ⎢
    ⎣
    −50 ·8 + 39 ·10
    −50 ·3 + 39 ·4
    −4 ·8 + 3 ·10
    −4 ·3 + 3 ·4
    		⎤
    ⎥
    ⎥
    ⎦

       ⎡
    ⎢
    ⎢
    ⎣
    −10
    6
    −2
    0
    		⎤
    ⎥
    ⎥
    ⎦    		    =    ⎡
    ⎢
    ⎢
    ⎣
    −10
    6
    −2
    0
    		⎤
    ⎥
    ⎥
    ⎦    		   
X = [
−50
39
−4
3
]
Find the determinant of A = [
3
2
0
−2
−3
5
7
4
1
].
  • To find the determinant of a matrix that is larger than 2×2, we must use minors and cofactors. Make sure to watch the lesson: it's much easier to understand this concept with a series of visual references. First, begin by choosing a horizontal or vertical line of entries from the matrix. For ease, let's just pick the top line to start with:
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    3
    2
    0
    −2
    −3
    5
    7
    4
    1
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢
    [Remember, absolute value bars around a matrix array means that we are taking its determinant, which is what we're doing in this case.]
  • For each of the entries in the line we chose, we pull out the entry, and multiply it by the determinant of the minor that entry creates. The minor is the matrix you'd get if you eliminated all the entries that are horizontally and vertically in line with the entry the minor is built around. For example, the minor for the 3 in the top-left corner is |
    −3
    5
    4
    1
    |. In addition, we must remember that when we pull out the entry, it becomes a cofactor, and the cofactors come with signs based on their location in the matrix. These signs are based on a checkerboard pattern of + and − as below:
    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    +
    −
    +
    −
    …
    −
    +
    −
    +
    …
    +
    −
    +
    −
    …
    −
    +
    −
    +
    …
    :
    :
    :
    :
    ···
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
    Thus, the top-left 3 would get a +, the top-middle 2 would get a −, and the top-right 0 would get a +.
  • Once we understand our cofactors and minors, we pull out each of the cofactors, multiply them by their associated minor, then sum it all up:
    3 ·⎢
    ⎢
    ⎢
    ⎢
    −3
    5
    4
    1
    		⎢
    ⎢
    ⎢
    ⎢    		    −2⎢
    ⎢
    ⎢
    ⎢
    −2
    5
    7
    1
    		⎢
    ⎢
    ⎢
    ⎢    		   + 0 ·⎢
    ⎢
    ⎢
    ⎢
    −2
    −3
    7
    4
    		⎢
    ⎢
    ⎢
    ⎢
    From before, we know how to take the determinant of a 2×2 matrix, it is
    ⎢
    ⎢
    ⎢
    ⎢
    a
    b
    c
    d
    		⎢
    ⎢
    ⎢
    ⎢    		    =    ad−bc,
    so going back to our determinant we get
    3 ·(−3·1 − 5 ·4)    −2(−2 ·1 − 5 ·7)   + 0
  • Continue to simplify, and that's the determinant:
    3 ·(−23)−2(−37)    =    −69 + 74    =    5
det(A) = 5
Find the value of  |
−8
0
4
−12
0
6
22
−4
17
|.
  • Remember, absolute value bars around a matrix array means that we are taking its determinant. To take the determinant of this matrix, we will use the method of cofactors and minors. Make sure you've watched the video lesson where you can see this method carefully explained and worked through. Also do the previous problem to this one first, as it works through things more gradually. Begin by identifying which horizontal or vertical line of entries you want to use from the matrix. You can choose to use any of them, and because this line will give us our cofactors, it helps to choose a line with a lot of 0's, since they'll cancel things out later on. With this in mind, let's choose the middle vertical line:
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    −8
    0
    4
    −12
    0
    6
    22
    −4
    17
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢
  • Pull these entries out as cofactors, and make sure you remember to apply the appropriate sign to each from the checkerboard pattern of signs:
    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    +
    −
    +
    −
    …
    −
    +
    −
    +
    …
    +
    −
    +
    −
    …
    −
    +
    −
    +
    …
    :
    :
    :
    :
    ···
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
    Thus the top 0 gets a −, the middle 0 gets a +, and the bottom −4 gets a −. With this in mind, pull out the entries as cofactors and multiply their associated minors:
    −0 ·⎢
    ⎢
    ⎢
    ⎢
    −12
    6
    22
    17
    		⎢
    ⎢
    ⎢
    ⎢    		    +0⎢
    ⎢
    ⎢
    ⎢
    −8
    4
    22
    17
    		⎢
    ⎢
    ⎢
    ⎢    		   − (−4) ·⎢
    ⎢
    ⎢
    ⎢
    −8
    4
    −12
    6
    		⎢
    ⎢
    ⎢
    ⎢
  • From there, simplify:
    −0 ·⎢
    ⎢
    ⎢
    ⎢
    −12
    6
    22
    17
    		⎢
    ⎢
    ⎢
    ⎢    		    +0⎢
    ⎢
    ⎢
    ⎢
    −8
    4
    22
    17
    		⎢
    ⎢
    ⎢
    ⎢    		   − (−4) ·⎢
    ⎢
    ⎢
    ⎢
    −8
    4
    −12
    6
    		⎢
    ⎢
    ⎢
    ⎢    		    =    4 ·⎢
    ⎢
    ⎢
    ⎢
    −8
    4
    −12
    6
    		⎢
    ⎢
    ⎢
    ⎢
    Remember, the determinant of a 2×2 matrix goes like this
    ⎢
    ⎢
    ⎢
    ⎢
    a
    b
    c
    d
    		⎢
    ⎢
    ⎢
    ⎢    		    =    ad−bc,
    so continuing to simplify, we get
    4 ·⎢
    ⎢
    ⎢
    ⎢
    −8
    4
    −12
    6
    		⎢
    ⎢
    ⎢
    ⎢    		    =    4 (−8 ·6 − 4 ·(−12) )     =     4 (0)     =     0
0
Find the determinant of  A = [
1
−3
0
2
0
6
0
−3
3
11
4
−1
0
4
−2
5
].
  • To take the determinant of this matrix, we will use the method of cofactors and minors. Make sure you've watched the video lesson where you can see this method carefully explained and worked through. Also do the previous two problems before this one, as they ramp up in difficulty, and this is fairly hard. Begin by identifying which horizontal or vertical line of entries you want to use from the matrix. You can use any of them, so since this will give us our cofactors, it helps to choose lines with a lot of 0's, since they'll cancel things out later on. With this in mind, let's choose the second horizontal line:
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    −3
    0
    2
    0
    6
    0
    −3
    3
    11
    4
    −1
    0
    4
    −2
    5
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
  • From there, we pull cofactors and minors. Make sure to apply the appropriate sign based on the location compared to our "sign checker board":
    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    +
    −
    +
    −
    …
    −
    +
    −
    +
    …
    +
    −
    +
    −
    …
    −
    +
    −
    +
    …
    :
    :
    :
    :
    ···
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
    For ease, we can immediately eliminate the minors attached to the 0 cofactors (since they will be reduced to 0 by multiplication), and so we won't even write them out:
    −0    +6 ·⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    0
    2
    3
    4
    −1
    0
    −2
    5
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		    − 0   + (−3) ·⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    −3
    0
    3
    11
    4
    0
    4
    −2
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢
    Thus, so far we have shown
    det
    		(A)     =     6 ·⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    0
    2
    3
    4
    −1
    0
    −2
    5
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		    −3 ·⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    −3
    0
    3
    11
    4
    0
    4
    −2
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢
  • At this point, we now need to find the determinants of each of those 3 ×3 matrices, so we'll tackle them one at a time. Using the top horizontal line for cofactors works fine for both, so we will do so.
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    0
    2
    3
    4
    −1
    0
    −2
    5
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		    =    1 ·⎢
    ⎢
    ⎢
    ⎢
    4
    −1
    −2
    5
    		⎢
    ⎢
    ⎢
    ⎢    		   −0    + 2 ·⎢
    ⎢
    ⎢
    ⎢
    3
    4
    0
    −2
    		⎢
    ⎢
    ⎢
    ⎢
    Then take the 2×2 determinants and simplify:
    1·(4·5 − (−1) ·(−2))   +   2 ·(3 ·(−2) + 4 ·0)     =     18 + 2 (−6)     =     6
    Thus, we have shown
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    0
    2
    3
    4
    −1
    0
    −2
    5
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		    =     6
  • We do the same process with the other 3×3 matrix:
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    −3
    0
    3
    11
    4
    0
    4
    −2
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		    =    1 ·⎢
    ⎢
    ⎢
    ⎢
    11
    4
    4
    −2
    		⎢
    ⎢
    ⎢
    ⎢    		   −(−3) ·⎢
    ⎢
    ⎢
    ⎢
    3
    4
    0
    −2
    		⎢
    ⎢
    ⎢
    ⎢    		   +0
    Then take the 2×2 determinants and simplify:
    (11·(−2)−4·4) + 3 (3 ·(−2) − 4 ·0)    =    −38 + 3 (−6)     =    −56
    Thus, we have shown:
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    −3
    0
    3
    11
    4
    0
    4
    −2
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		    =    −56
  • To summarize, we have found:
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    0
    2
    3
    4
    −1
    0
    −2
    5
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		 = 6          ⎢
    ⎢    		          ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    −3
    0
    3
    11
    4
    0
    4
    −2
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		 = −56
    We know
    det
    		(A)     =     6 ·⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    0
    2
    3
    4
    −1
    0
    −2
    5
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		    −3 ·⎢
    ⎢
    ⎢
    ⎢
    ⎢
    1
    −3
    0
    3
    11
    4
    0
    4
    −2
    		⎢
    ⎢
    ⎢
    ⎢
    ⎢    		 ,
    so we plug in to find det(A):
    det
    		(A)     =     6 ·(6) − 3·(−56)     =     36 + 168     =     204
det(A)=204
Find the inverse to  A=[
1
2
−4
1
3
−2
−2
0
12
].
  • Note: The method to find matrix inverses that are larger than 2×2 requires you to be familiar with augmented matrices, row operations, and Gauss-Jordan Elimination. Make sure you already know all of these things, which can be found at the beginning of the next video lesson. Furthermore, make sure you have watched the final portion of the video for the current lesson you are on (Determinants and Inverses of Matrices) as it explains how the method is used and gives a simple example.
  • For an n×n matrix A (whose determinant is not 0), we find the inverse by first creating an augmented matrix with the identity matrix:
    ⎡
    ⎣
    A
    In
    		⎤
    ⎦
    Then, through the process of Gauss-Joran elimination and by using row operations, we make the left half of the augmented matrix In, and we end up with
    ⎡
    ⎣
    In
    A−1
    		⎤
    ⎦    		 ,
    where the inverse matrix has appeared on the right side.
  • Following this method, begin by turning A into an augmented matrix with In on the right side:
    ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    −4
    1
    3
    −2
    −2
    0
    12
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦    		    ⇒    ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    −4
    1
    0
    0
    1
    3
    −2
    0
    1
    0
    −2
    0
    12
    0
    0
    1
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
  • Now that we have an augmented matrix, we begin to apply row operations through Gauss-Jordan elimination. To begin with, below the main diagonal we cancel out to 0's and put the main diagonal as 1's:
    ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    −4
    1
    0
    0
    1
    3
    −2
    0
    1
    0
    −2
    0
    12
    0
    0
    1
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    −R1 + R2 →
    2R1 + R3 →
    		⎡
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    −4
    1
    0
    0
    0
    1
    2
    −1
    1
    0
    0
    4
    4
    2
    0
    1
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    −4R2 + R3 →
    		⎡
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    −4
    1
    0
    0
    0
    1
    2
    −1
    1
    0
    0
    0
    −4
    6
    −4
    1
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    − 1
    4
    		·R3 →
    		⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    −4
    1
    0
    0
    0
    1
    2
    −1
    1
    0
    0
    0
    1
    − 3
    2
    1
    − 1
    4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
  • From there, we now cancel above the main diagonal to 0's, working our way back up:
    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    −4
    1
    0
    0
    0
    1
    2
    −1
    1
    0
    0
    0
    1
    − 3
    2
    1
    − 1
    4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
                            
    4R3+R1 →
    −2R3+R2 →
    		⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    0
    −5
    4
    −1
    0
    1
    0
    2
    −1
    1
    2
    0
    0
    1
    − 3
    2
    1
    − 1
    4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
                            
    −2R2+R1
    		⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    1
    0
    0
    −9
    6
    −2
    0
    1
    0
    2
    −1
    1
    2
    0
    0
    1
    − 3
    2
    1
    − 1
    4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
  • At this point, our augmented matrix now has an identity matrix on the left side, so the right side is the inverse to our original matrix:
    A−1     =    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    −9
    6
    −2
    2
    −1
    1
    2
    − 3
    2
    1
    − 1
    4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦
    After this much effort, it's a good idea to check our work, so let's make sure that A A−1 comes out to be the identity matrix, like it should:
    A A−1 =⎡
    ⎢
    ⎢
    ⎢
    ⎣
    1
    2
    −4
    1
    3
    −2
    −2
    0
    12
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦    		⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    −9
    6
    −2
    2
    −1
    1
    2
    − 3
    2
    1
    − 1
    4
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦

    ⎡
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎢
    ⎣
    1·(−9) + 2 ·2 −4 ·(− 3
    2
    		 )    
    1 ·6 + 2 ·(−1) −4 ·1
        1 ·(−2) + 2 · 1
    2
    		− 4 ·(− 1
    4
    		 )
    1·(−9) + 3 ·2 −2 ·(− 3
    2
    		 )    
    1 ·6 + 3 ·(−1) −2 ·1
        1 ·(−2) + 3 · 1
    2
    		− 2 ·(− 1
    4
    		 )
    −2·(−9) + 0 ·2 +12 ·(− 3
    2
    		 )    
    −2 ·6 + 0 ·(−1) +12 ·1
        −2 ·(−2) + 0 · 1
    2
    		 +12 ·(− 1
    4
    		 )
    		⎤
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎥
    ⎦

       ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    1
    0
    0
    0
    1
    0
    0
    0
    1
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦    		   
A−1     =     [
−9
6
−2
2
−1
[1/2]
−[3/2]
1
−[1/4]
]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Determinants & Inverses of Matrices

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:06
  • Not All Matrices Are Invertible 1:30
    • What Must a Matrix Have to Be Invertible?
  • Determinant 2:32
    • The Determinant is a Real Number Associated With a Square Matrix
    • If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
  • Determinant of a 2 x 2 Matrix 4:34
    • Think in Terms of Diagonals
  • Minors and Cofactors - Minors 6:24
    • Example
  • Minors and Cofactors - Cofactors 8:00
    • Cofactor is Closely Based on the Minor
    • Alternating Sign Pattern
  • Determinant of Larger Matrices 10:56
    • Example
  • Alternative Method for 3x3 Matrices 16:46
    • Not Recommended
  • Inverse of a 2 x 2 Matrix 19:02
  • Inverse of Larger Matrices 20:00
  • Using Inverse Matrices 21:06
    • When Multiplied Together, They Create the Identity Matrix
  • Example 1 23:45
  • Example 2 27:21
  • Example 3 32:49
  • Example 4 36:27
  • Finding the Inverse of Larger Matrices 41:59
  • General Inverse Method - Step 1 43:25
  • General Inverse Method - Step 2 43:27
    • General Inverse Method - Step 2, cont.
  • General Inverse Method - Step 3 45:15

Transcription: Determinants & Inverses of Matrices

Hi--welcome back to Educator.com.0000

Today, we are going to talk about determinants and the inverses of matrices.0002

Consider if we wanted to find x in the equation 5x = 10--pretty basic algebra, right?0006

We would cancel out the 5 by dividing it on both sides.0012

Or equivalently, we could think of this as multiplying by 5 inverse, which is just 1/5.0014

If we multiply by 1/5 on both sides, we cancel out the 5, because the multiplicative inverse to 5 is 1/5; that why it is 5-1.0020

What if we wanted to solve for the matrix X in the equation below?0028

We had some matrix A, times the matrix X, is equal to the matrix B.0031

We have this matrix equation; so we need to somehow cancel out A to get X alone.0036

It is the same basic idea; we just need to cancel out an entire matrix.0041

So, we need to multiply both sides by the inverse to A; this means we need to find the inverse to A.0044

If we can find this magical inverse, then we could multiply both sides.0050

We would have A-1AX and A-1B.0054

Well, the A-1 and the A will cancel each other out, and we would be left with X = A-1B.0058

So, we would be able to solve for that matrix, that unknown matrix, X, if we wanted to,0064

in terms of this A-1 and B, if we know what A and B are.0068

It is very similar to 5x = 10; we multiply by the multiplicative inverse of 5, 5-1, on both sides, to get what x is.0072

So, AX = B...we multiply by the multiplicative inverse of A on both sides to get that X alone.0079

Not all matrices are invertible; consider if we wanted to solve for X in the basic equation 0x = 0.0086

It would be impossible: the information about what x is has just been destroyed by that 0.0094

0 multiplied by anything is going to come out to be 0, so we don't have any idea what that x is anymore.0099

There is no way to cancel out 0, because 0-1 does not exist.0104

There are some special things out there that we can't invert.0109

There is no way to flip them to an inverse, because 0...you can't invert it.0112

You can't reverse the process of multiplying by 0; it is gone--the information is lost.0117

It is the same thing going for matrices: not all matrices can be inverted.0123

A matrix that can be is called invertible: if we can invert a matrix, we call it invertible, or we might call it non-singular.0127

If a matrix cannot be inverted, it is called singular.0134

To be invertible, a matrix must have two properties: the matrix must be square--it has to be a square matrix to invert;0138

and the determinant of the matrix must be non-zero.0145

So, what is a determinant? Let's start talking about determinants.0149

The determinant is a real number associated with a square matrix.0152

The determinant of a matrix A is denoted by either detA (like determinant of A--we are shortening it),0156

or vertical bars on either side of the matrix A.0163

Now, A may look similar to absolute value, but it is not; it is not absolute value--it is the determinant of A.0165

So, when it is vertical bars around a matrix, we are talking about determinant, not absolute value.0172

So, vertical bars around a matrix, unlike absolute value, can produce any real number, including negative numbers or 0 or positive numbers.0176

So, it is not limited to just giving out positive or 0, like absolute value; it is allowed to put out anything.0186

So, don't get confused by those vertical bars, thinking that that implies positiveness; it doesn't.0191

For the most part, though, I prefer this detA thing, this determinant of A; so that is the form that we will be seeing.0195

But occasionally, you will see it with the vertical bars, instead.0202

The determinant of a matrix has many important applications and properties.0204

There is a huge amount of stuff that this determinant is useful for.0207

But we are not going to get into that in this course.0211

In this course, we are only going to concern ourselves with one thing: whether or not a matrix is invertible, and the fact that a determinant tells us that.0213

If a determinant of a matrix is non-zero, then the matrix is invertible, and vice versa.0220

So, if the determinant of A is not equal to 0, then we know that A is invertible.0228

And if A is invertible, then we know that the determinant of A must not be equal to 0.0232

On the flip side, if the determinant of A is equal to 0, then we know that A is not invertible.0236

And if A is not invertible, we know that the determinant of A is equal to 0.0241

So, just remember that detA not equal to 0 means that it is invertible.0245

And that really works a lot like we are used to with the real numbers.0249

You can invert any number you want, except 0.0253

It is the same thing with matrices: you can invert any matrix you want, except for ones that have determinant 0.0256

All right, so think in terms of that: detA not 0 means invertible--you are allowed to invert; detA = 0--you are not allowed to invert.0261

So, let's see the determinant of a 2 x 2 matrix: if A = a, b, c, d, it is given by detA,0271

which is equal to this other way to write determinant of A, comes out to be ad - bc.0279

A good mnemonic to remember this is to think in terms of diagonals--0286

the down diagonal, ad, multiplied together, and then subtracted by this up diagonal here, cb or bc, so minus bc.0289

We subtract by that up diagonal.0303

Let's look at an example--let's do an example here: Multiply...0305

if we want to take the determinant of 5, 9, 3, 4, notice that we have these bars on either side.0309

If we have bars of some matrix inside, what that is saying is to take the determinant of that stuff on the inside.0315

Bars on either side is just like the bars on either side of the capital letter denoting the matrix.0326

It says to take the determinant of whatever is inside of there.0332

So, you will see that notation a lot; but when we are talking about just letters, I prefer that one.0334

OK, in either case, if we are taking the determinant of the matrix 5, 9, 3, 4--if we are taking this one right here,0339

the determinant of 5, 9, 3, 4, the first thing we do is take the down diagonal.0344

So, it is going to be 5 times 4; and then, it is going to be minus the up diagonal, 3 times 9, so 9 times 3.0351

5 times 4 - 9 times 3; we get 20 - 27, and that comes out to be -7.0361

Once again, the determinant can come out to be any number; it doesn't have to come out to a positive;0372

it just has to come out to any real number at all.0376

Minors and cofactors: first we are going to talk about minors.0380

Before we can look at determinants of larger matrices, we will need two concepts: minors and cofactors.0383

First, we are going to look at minors.0388

For a square matrix A, the minor, mi,j (remember, i is the row i; j is the column j)0391

of the entry ai,j is the determinant of the matrix obtained by deleting the ith row and jth column.0398

So, we go to this i,j location, this ai,j entry, and we delete out from that, vertically and horizontally.0405

So, we will take some location, and then we will delete out horizontally, delete out vertically, and group back together and see what is left.0414

For example, if we have a below, we would have m2,3; 2,3 means we are on the second row, and we are going to be on the third column.0421

We are looking at -8 as the epicenter of where this thing is.0431

That is the entry ai,j; so the entry 2,3 would be -8.0436

Now, we delete (the determinant of the matrix obtained by deleting) the ith row and jth column.0440

We delete this second row; we delete this third column; and we see what the matrix is that is left.0446

Well, the matrix that is left is 6, 2, -7, 3; that is all that hasn't been crossed out.0453

Now, we go and we take the determinant of that; we are taking the determinant with these bars,0461

because the minor is that you delete, and then you take the determinant.0466

So then, we just take 6 times 3, minus -7 times 2; 18 + 14...we get 32, so that is our minor.0469

Cofactor is very closely based on the minor.0480

The cofactor just multiplies the minor by 1 or -1, based on the location of the entry the minor comes from.0484

So, there is this shifting, flipping-back-and-forth pattern of positive/negative that is really deeply connected to the determinants of matrices.0491

The cofactor ci,j, the ith row, jth column cofactor, of the entry ai,j,0500

the entry in the ith row, jth column of our matrix A, is given by ci,j0505

is equal to -1 to the i + j times that minor i,j.0511

So, the -1 to the i + j is just a way of saying if it is going to be positive or negative.0517

-1 to the 0 is positive; -1 to the 1 is negative; -1 to the 2 is positive; -1 to the 3 is negative; -1 to the 4 is positive.0521

-1 to the even number is positive; -1 to the odd number is negative.0529

So, we can see this as an alternating sine pattern.0533

If we are in -1, to the row 1 + column 1, then that is going to be -1 squared; -1 squared comes out to be positive 1.0535

There we are at row 1, column 1.0546

If we were to instead, say, look at row 2, column 3, then it would be -1 to the 2 + 3, which is equal to -1 to the 5.0549

So, since it is to an odd number, it is going to be negative; so we get that negative there.0561

We can see this in terms of the i + j thing; but we can also see it in terms of this alternating sign pattern.0566

I would recommend, any time you are working with cofactors, that you just draw up the alternating sign factor to whatever size you are doing.0572

For example, if you are working with a 3 x 3, just draw out a 3 x 3 alternating sign pattern.0578

It always starts with a positive in the top left: so + - +, - + -, + - +.0584

And then, from there, you will be able to work from it and use that as a reference point; we will see that in the examples.0593

Thus, based on our previous example, when we took what m2,3 was (m2,3 was equal to0598

the determinant of 6, 2, -7, 3, because c2,3 is still going to be based around row 2, column 3;0605

so -8...cross out...cross out...6, 2, -7, 3...the same thing here; and then we just take the determinant of that);0612

but we are here in the 2,3 position in our alternating sign pattern (or alternately, if we want to look at it0621

in terms of -1 to the i + j--either way would end up working out the same); so we have this negative here,0629

so we have a negative showing up here, so that will end up coming out to -32,0636

because we already figured out that the determinant for that minor is 6, 2, -7, 3; that is what we get out of that.0641

And so, that came out to be positive 32; so when we have this sign on top of that, that is going to come out to be -32.0648

All right, so how do we actually take the determinant?0656

Let's apply this stuff: the determinant of an m x n matrix A is given by the sum of the entries in any row or column0657

(you can choose any row or any column at all), and you multiply each one of those entries0665

by the respective cofactor that would come out of that entry.0670

So, the determinant of A, which is equal to another way to say the determinant of A,0674

is equal to...say we chose the kth row; then we would have ak,1,0677

the first entry in the kth row, times the cofactor of the kth row, first entry,0681

plus ak,2, the kth row, second entry, times the cofactor for the kth row, second entry,0687

up until the kth row, nth entry, and kth row, nth entry cofactor.0694

Similarly, we could have also done this with columns; it would be the first entry, kth column of A,0701

times the cofactor for the first entry, kth column; or the second entry, kth column,0706

with the cofactor of second entry, kth column, up until the nth entry, kth column,0711

nth entry, kth column cofactor.0715

So, that is how it works; don't worry--we will see an example that will make this make a lot more sense.0718

Note that this is true for any value of k, as long as 1 ≤ k ≤ n.0722

So, our k has to be somewhere in these m x n; we can't choose a row that is beyond the dimension,0728

or a column that is beyond the size, of our matrix; that doesn't make sense.0733

But as long as we choose a row that is inside of our matrix, and a column that is inside of our matrix, we can choose any one at all.0737

So, this process can be done with any row or any column, and you will end up getting the exact same result--kind of amazing.0742

We won't see why, but it is pretty cool.0748

This usually means that it is in our interest to choose the row or column that has the most zeroes,0750

because it is really easy: 0 times a cofactor--we don't have to worry about what the cofactor is.0755

It is just immediately going to eliminate itself.0761

So, the thing with the most zeroes, the row or column that has the most zeroes (or the smallest numbers,0763

if we don't have that many zeroes) will help make calculation easier; so that is something to stay on the lookout for.0768

All right, determinant of larger matrices: let's actually put "rubber to the road" and see how this works.0775

First, we notice that there is a 0 here; I like going horizontally, so let's work out this way.0782

Now, notice: a 3 x 3 sign pattern (put it inside of vertical lines, just so we are reminded that we are doing a determinant)0787

is going to look like this: so let's work on this horizontal line here.0802

The first entry in this row is 1; we then go out; we cross out the things on a line with that.0809

That would bring us to the entry 1, times the sign for that cofactor, so -1, times the minor, 2, 3, 3, 5.0819

Next, it is going to be a +; our next one in the row is going to be the 0.0836

We cut out...we don't even really have to care about the cutting out, because 0 times whatever0843

we end up having inside for that minor--that is going to get knocked out, so it doesn't really matter.0849

That is the beauty of choosing the 0.0853

Next, we have the -8; so -8 knocks out what is there; -8, and we are on this one, so - -8, times the minor0855

that is produced by cutting around that -8, cutting a vertical and a horizontal on that -8: 6, 2, -7, 3.0867

We work these out; we have -1 times...down diagonal, 2 times 5, minus up diagonal; 10 - 9 becomes -1.0876

Minus, plus...so that becomes + 8 times...6 times 3 becomes 18, minus...-7 times 2 is -14; that cancels out.0889

So, we have -1, negative and negative; that becomes positive 1...0904

Oh, sorry, that did not become negative back here: 2 times 5 is 10, minus 3 times 3 is 9, so we have 10 - 9 is 1; sorry about that.0911

So, that should have been a 1; here is a -1, so this comes out to be -1; it does not cancel out; I'm sorry about that.0921

Then, + 8 times...18 + 14 becomes 32, so -1 +...8 times 32 is 256; so we end up getting 255 as the determinant for this matrix.0928

Alternatively, we could have chosen a different row or a different column.0946

For example, we could have just gone along the top, like this, and we would have had 6 times...0949

and it would be positive, if we are going along the top there...so 6 times...we cross out around it; 0, -8, 3, 5.0955

And then, the next one is minus, -2, times...the minor around that 2 here would cross out to be 1, -8, -7, 5.0964

And then finally, + (because it is a plus in our signs) 3 times 1, 0, -7, 3.0976

You can work it out that way, as well, and you would end up getting 255, as well.0985

I like this row here, because we had that 0; and so, it just managed to knock itself out, right from the beginning.0989

That is that much less calculation for us to have to deal with; I think that is nice--less calculation makes it easier.0995

All right, there is an alternate method for finding the determinant of a 3 x 3 matrix that some people teach.1001

Personally, I want to recommend against using this method--I don't really think there is a good reason to use it.1008

The method we just did, that method with the cofactor expansion, while it seems a little complex at first1014

(it is a lot of things going on) will work for any size matrix at all.1019

And to be honest, this alternate method doesn't actually go any faster, I don't think.1023

So, I would say to try to stick to the cofactor method; I think it works better in general.1029

It gives you the ability to cancel out a whole bunch of zeroes, if you see a bunch of zeroes.1034

And you can use that same method for any size matrix and work down to smaller things.1038

That said, you might have to know it for class, or you might just really want to use it.1043

So, if you must know it, here it is.1046

The first thing you do: you begin by taking the first two columns of the matrix, and you repeat them on the right of the array.1049

So, we have 6, 2, 3, 1, 0, -8, -7, 3, 5; that shows up here, just like normal.1054

But then, we take the first two columns, and then we also repeat this on the right side.1062

So now, we have this extra-large array of numbers.1069

Once we have that array of numbers, we can work with it.1073

We multiply each red down diagonal--we multiply these together, and we add them up.1076

In this case, we would have 6 times 0 times 5; 2 times -8 times -7; and 3 times 1 times 3; that is what we get out of there.1082

And then, we subtract by each of the up diagonals, those blue ones, multiplied together; you subtract by those.1091

Minus (it is always going to be minus), and then -7 times 0 times 3, and then minus1098

(we are subtracting again) 3 times -8 times 6, and then minus 5 times 1 times 2.1105

You work that all out and do a bunch of calculation; you end up getting the exact same number, 255.1112

So, it is an alternate way to find the determinant; it will work if you have a 3 x 3 matrix.1116

It is not that bad; but I don't really think there is a whole lot of reason to use it.1121

It doesn't really go that much faster; you basically have to deal with the same amount of arithmetic.1125

And it is a very specific trick for something that you might have to do on a larger scale, and you can't use that trick anymore.1129

So, I would recommend using the method we were just talking about, with cofactors and minors.1135

But if you really want to use this one, here it is.1139

All right, we are ready to finally see the inverse of a 2 x 2 matrix.1142

So, if we have some 2 x 2 matrix, A = a, b, c, d; then the inverse of A, assuming that the determinant of A is not equal to 0,1145

(if the determinant of A is equal to 0, then we can't invert it at all), then A-1 = 1/ad - bc, times the matrix d, -b, -c, a.1152

So, notice: what we have done there is flipped the location of the diagonal here, and then we put negatives on the b and the c.1168

That is what we are getting here and here and here and here.1176

That is one way of looking at what is going on.1181

Equivalently, you could also write this as 1/detA, because the determinant of A is just ad - bc.1183

So, detA is the exact same thing; and then we are going to end up having the same matrix here and here.1190

That is another way to think about it and remember it; that might be a little bit easier.1195

All right, for the most part, at this level of this course or any similar math class,1199

you are probably not going to need to compute the inverse of a matrix that is any larger than a 2 x 2.1203

You are almost certainly not going to need to do that by hand.1208

But your teacher might want you to; you might just be curious about it.1211

So, if for some reason you need to calculate the inverse of a matrix that is larger than a 2 x 2 matrix,1214

and you have to do it by hand, we will go over a method for this after the examples.1219

We will talk about that after the examples; we will see something for doing that.1223

There is...notice, I said "by hand"; it turns out that if you have a graphing calculator (or access to the Internet),1227

you can actually just plug in matrices and have other computers invert them for you.1233

It is a very useful thing, because the arithmetic of it is very simple, but tedious, and there is a lot of arithmetic.1238

So, we will talk about that a little bit more in the next lesson.1244

Or we will talk about how there are calculators and matrices interacting together.1247

But that is something to think about, if you have to take the inverse of a matrix that is really large;1250

but you can not do it by hand--you are not required to show all of your work by hand--you might want to just use a calculator.1255

That is something to think about.1262

All right, how do you use inverse matrices?1264

If you have some A and A-1, then we know that A-1 times A times B equals B.1266

A-1 and A cancel each other out, and they have no net effect.1272

This is because A-1 times A equals the identity matrix, which is equal to A times A-1.1276

So, if you have the inverse to a matrix, you can multiply on the left side or the right side, and it will create the identity matrix.1281

It creates the identity matrix I, which as we noted in the previous lesson has no effect in multiplication.1287

A-1A up here becomes I; and then I times B--well, the identity matrix times anything becomes just what we already had.1294

So, we get B; so that is why A-1 and A are cancelling out.1301

They turn into the identity matrix, and then that just doesn't do anything.1304

I want you to notice that we can multiply from the left side or the right side.1307

It doesn't matter; it will cancel out in either direction.1311

That is one of the nice things about inverses; they actually will commute, unlike pretty much everything else with matrices.1313

It is important to note that, if we multiply an equation by a matrix on both sides, we have to choose a direction to multiply from1319

and do the same for both parts of the equation.1326

So, if we multiply from the left, we have to multiply from the left on both sides.1329

If we multiply from the right, we have to multiply from the right on both sides.1333

This is because pq is not equal to qp, in general.1336

Multiplying on the left by p is generally very different than multiplying on the right by p.1340

So, if we are going to keep up equality, we have to do the same action; we have to multiply from the left on both sides,1344

because multiplying from different sides is actually a different action with matrices.1349

So, you have to make sure that you multiply from the same side if you want to keep the equality of the equations.1353

So, for example, if we have that A = B, then we can have CA = CB, where we multiply on the left for both sides.1358

Or we could have AC = BC, where we multiply on the right for both sides.1366

But usually, in general, CA is not going to be equal to BC, where we multiply on the left for one, and we multiply on the right for the other.1372

It is in general not going to end up being true; so you will have lost your equality.1380

So, make sure you notice that sort of thing; be careful here--it is dangerous.1384

It is really easy to make this mistake, because so often, when we think about multiplying numbers and equations,1390

like x = 10...we might multiply 3x = 10 times 3, but that is not how it can work in matrices.1395

The only reason we can get away with that in a normal equation is because they commute, so it doesn't matter which side we multiply from.1400

But with matrices, it matters which side we multiply from; so we can't have CA = BC; we have to make sure it is either CA = CB or AC = BC.1408

We have to make sure that we are multiplying both on the left or both on the right.1417

All right, we are ready for some examples.1421

What is the determinant of this 3 x 3 matrix? -2, 1, -3, 4, 2, 0, -1, 0, 1.1423

Our very first thing that we want to do is make a sign marker, just so we can see where all of the signs show up.1429

So, at this point, we need to choose some row or some column to work with.1440

We could choose the top one; that would be fine, but it doesn't have any 0's in it.1446

It has some numbers that are larger than that; so I like this one, because it has -1, 0, and 1.1449

So, one of them is going to cancel out, and the other ones have very little effect on the numbers.1454

Let's work with that: -1 will cancel out those; so we have -1 times 1, -3, 2, 0.1458

Then, the next one...it is still that, because that corresponds to that sign right there...1472

Next, we have minus, because it corresponds to that one, 0, times...and we could figure out what this is,1480

but it doesn't matter; because it is 0, it is going to knock itself out automatically.1490

0 times anything is going to come out as 0, so we don't even have to worry about computing it.1494

And then finally, the 1: that will knock out these, so we have a + here, + 1, times -2, 1, 4, 2.1498

We calculate this; we have -1 times...1 times 0 is 0; 2 times -3 is -6; but it is minus that;1512

so 1 times 0 is 0; minus 2 times -3, so it is a total of +6.1521

And then, plus...1 times...just figure out what this is...-2 times 2 is -4; minus 4 times 1, another -4; so we have a total of -8.1529

We work this out; we have -6 - 8; and we get -14.1540

There are many ways to have done this; we could have also chosen to do this based on this column here.1547

Really quickly, we would have had -3, since we are starting here.1551

We start at positive, but it starts at -3; so -3 times 4, 2, -1, 0,1557

minus...our next sign...0 times...we don't even have to care about it, because it will just knock itself out... plus 1 times -2, 1, 4, 2.1564

Or we could have gone from a different place entirely.1576

We could have also had this, and this would be equal; all of these ways will end up coming out to be the exact same thing.1579

That is one of the cool properties of the determinant.1583

-2 times 2, 0, 0, 1, minus 1 times 4, 0, -1, 1, plus -3 times 4, 2, -1, 0.1587

There are many different ways to do this: this here is the same as this here, is the same as this here.1606

They all end up being equal to -14; so the question of how we want to approach this--1614

which row, which column--we just choose whichever one seems easiest to us.1620

And even if we end up choosing the wrong one--we choose one that is slightly harder--1624

it doesn't matter, because they all come out to be the same thing.1627

We might have to do a few more extra arithmetic steps, but in the end, we will still get the same answer; so it is OK.1629

You don't have to really worry about that.1634

All right, what is this one? We have a 4 x 4, so at this point, we have to take the determinant of this.1636

The first thing we want to do is get a nice sign grid, so we can see all of our plusses and minuses.1641

+ - +...always a positive in the top left...- + - +, + - + -, - + - +; great.1646

So, at this point we want to figure out which is our best row or column to choose.1656

I see two zeroes on this column; so to me, that looks like it is going to make it easiest; I am going to go with that one.1660

I have the 2; it crosses out these; that corresponds to this +1 here, so I just have 2 times...1666

I cross out those other ones; I am left with -1, 3, 0, -4, 5, 4, 1, 1, 0.1676

OK, and then 0 here and the 0 here...we don't even have to worry about them,1684

because they are just going to multiply out to cancel out entirely.1689

So, we only get to having to worry about the 3; that leaves us here.1692

So, it is minus 3 times what gets crossed out: 3 times 2, -1, 3, 0, -4, 5, -3, 1, 1.1695

OK, at this point, let's figure out, of these new ones, which ones we want to use.1714

Let's make a new, smaller, 3 x 3 sign grid, so we can think in terms of that now.1719

OK, so this one...what seems easiest to me is this column...and I would say this row here.1726

We will work with those: we have 2 times whatever the determinant of that larger 3 x 3 is (this one right here);1732

we are working with the 0, so the 0 is going to just knock things out;1741

the only one that we really have to care about is this 4; it will be 4 times...1744

oh, wait, 4 here is there; so we have a -4; we always have to pay attention to that cofactor1750

bringing either a plus or a negative; that is why we make these sign grids here and here.1755

So, we have to pay attention to cofactors.1760

-4 times...that would cross out these things, so...-1, 3, 1, 1.1762

And then, over here, minus 3; so we chose this one, so we are going to have this row starting here: -3 times -0...1773

you don't have to worry about that one; plus...-4 times...it crosses out the other ones1782

that it is horizontal and vertical on...2, 3, -3, 1 is what is left there;1791

And then, minus 5...it crosses out, and we get 2, -1, -3, 1.1799

All right, we start working these out; since they are 2 x 2 matrices, we can just work them out now.1812

So, we have 2 times -4 times...-1 times 1 is -1, minus 1 times 3 is -4.1816

Then, minus 3 times -4 times 2 times 1 (is 2), minus -3 times 3; so 2 - -9 gets +11.1826

2 - 3(3)...we have -9...so -4 times 11, minus 5 times...2 times 1 is 2, minus 3(-1)...so 2 here, and then minus 3...1846

-3 times -1 becomes positive 3, but we are subtracting by that, so it is 2 minus 1...2 minus 3...so we get -1.1868

OK, so keep working that out: 2 times -4 times -4 is going to come out to be 2 times +16...1882

minus 3...-4 times 11 is -44; these cancel out, and we get + 5; 2 times 16 is 32, minus 3...-44 + 5 is -39.1893

These negatives cancel out; at this point we have this equal to 32 + 3(39) is going to be the same as 3(40) - 3, so + 117.1915

32 + 117 comes out to be 149; so the determinant of our matrix is equal to 149.1929

Great; so by carefully choosing which row we decide to work with, we can make this a whole lot easier.1942

By choosing that third row down, we were able to get a 0 to show here and a 0 to show here,1947

which allowed us to cancel out all of the things, so we only had to figure out two 3 x 3 determinants,1953

which is a lot easier than having to figure out four of them or more--anything like that.1957

So, by carefully choosing the row or column that you do your cofactor expansion on, you can make things a lot easier on yourself.1961

The third example: Prove that, for any 2 x 2 matrix A, where the detA is not equal to 0,1967

then A-1 = 1/(ad - bc) times the matrix d, -b, -c, a.1972

One thing that should be written here is that A is going to be equal to our standard form for just writing a general one, a, b, c, d.1977

So, how would we prove this? Well, we just prove it by showing that A times this supposed A-11985

does, indeed, come out to be the identity matrix, because that is what it means to be the inverse.1991

That is that something times its inverse comes out to be the identity matrix.1995

Some matrix times its inverse matrix comes out to be the identity matrix.1999

That is what it means to be an inverse for matrices.2002

So, let's just check that: let's say A-1 times A.2005

We don't know for sure that it actually is going to turn out to be the inverse, but let's try it.2011

We were told that the detA is not equal to 0; it is the determinant of A...2015

Well, remember: if this is our A right here, then the determinant of A is going to be equal to ad - bc.2019

This would be our only worry in creating this A-1: 1/(ad - bc)--if it is dividing by 0, everything blows up.2026

But since we are told that the determinant of A (which is equal to ad - bc) is not equal to 0,2033

we know that we don't have to worry about dividing by 0, so we can move on.2037

A-1 times A: we have 1/(ad - bc), times the matrix d, -b, -c, a.2040

And then, times A is a, b, c, d...so first, we work through with matrix multiplication.2053

We have our 1/(ad - bc); we will scale later; right now, it will be easier to just work with just the variables, without that fraction getting in the way.2063

So, the first column: we know we will get out to a 2 x 2 matrix in the end; so first row times first column:2074

d times a...actually, let's expand this even more...minus b times c; great.2083

The next one: d times b, minus b times d.2093

The second row on the first column now: -c,a on a,c; -c on a gets us -ca; a on c gets us + ac.2102

The last one: -c,a on b,d gets us -cb + ad.2111

So, we see this; and we do a little bit of simplification, moving things around.2119

Well, db - bd...since b and d are just real numbers, they are commutative, so db - bd just cancel each other out.2124

-ca + ac: once again, they knock each other out.2134

We can rearrange things a little bit; so we have 1/(ad - bc) times the matrix.2137

Well, da - bc is the same thing as ad - bc; this is 0, and this is 0; and -cb + ad...well, we can write that as ad - bc.2142

So, 1/(ad - bc) times this...well, we will get 1, 0, 0, 1, which is exactly what we were looking for.2153

So, this is, indeed, equal to identity matrix; and if we were to do it the other way,2162

A times A-1, to multiply our inverse from the right side, it would end up coming out the same; we would get the same answer.2166

And it turns out that, if you find a matrix that works on one side, you know that it has to work on the other side.2173

But that is a little bit of a deeper result that we haven't talked about explicitly.2178

But you could prove this just by hand, if you wanted to show AA-1; but that is pretty good.2181

The final example: Given that B = -2, 3, 0, 4, and AB = -6, 29, 4, 22, find the matrix A.2186

How are we going to do this? We don't know what A is.2194

We know what AB is; we know what B is; well, notice that we can create a plan like this:2196

AB = AB--that is kind of obvious, but it is true.2201

So, if we came along, we could knock out that B with B-1, so we could have AB = AB,2206

and then we would come along and hit it with B-1 on both sides.2216

And now, we could rewrite this as A =...well, we could cancel out to A on the right side,2221

but we could also see that it is just AB times B-1.2227

We know AB; we know B; and so, if there is a B-1, we can figure out what it is from our B.2235

So, our first step is to figure out what B-1 is.2243

And then, once we know what B-1 is, we just have AB times B-1, and we will have our A.2246

So, that is our theoretical understanding; now it is time to just do the arithmetic.2252

If B = -2, 3, 0, 4, then B-1 equals 1 over the determinant, which is ad - bc,2256

so -2 times 4, minus 0 times 3; so that is -8; times...we flip the location of the main diagonal,2265

and then we put negatives on the other ones: -3 and -0 (we can write as just 0).2274

Simplify that just a little bit to -1/8 times 4, -3, 0, -2.2279

Great; so at this point, we know, from what we showed here, that A is equal to AB times B-1.2287

Well, we know that AB is -6, 29, 4, 22; and B-1 is -1/8 times 4, -3, 0, -2.2299

So, I think it is easier to bring the fraction in afterwards; so let's pull the fraction to the front.2320

The fraction there is just a scalar, so it is just going to scale the matrix.2324

We can scale the matrix any time we want; let's just pull it out to the front, so we can have our matrices do their multiplication.2328

We have -6, 29, 4, 22; 4, -3, 0, -2; OK.2334

There is still that fraction up at the front: -1/8 times whatever comes out of this.2348

It will come out to be a 2 x 2; -6, 29 times 4, 0: -6 times 4 gets us -24; 29 times 0 is just 0.2354

-6, 29 on -3, -2; -6 times -3 gets us positive 18; 29 times -2 gets us -58; so that gets us a positive 40.2368

Oops, I'm sorry; it is not positive; 29 times -2 got us -58, so it is a -40; I'm sorry about that.2382

4, 22 on 4, 0: 4 times 4 gets us 16; and 22 times 0 is just 0.2388

4, 22 on -3, -2: 4 times -3 is -12; 22 times -2 is -44; -12 - 44 comes out to be -56.2396

So, at this point, we can use our -1/8; we simplify this out: we get -1/8 times -24 will become...24/8 is 3; the negatives cancel out, so we get +3.2407

-1/8 times -40 becomes positive 5; -1/8 times 16 becomes -2; -1/8 times -56 becomes positive 7.2419

We have A = 3, 5, -2, 7; and there we are.2429

We had to do a lot of arithmetic to get to this point, so let's double-check and make sure that that is the answer.2436

We know that A times B has to be this right here, because we were given AB, right from the beginning.2441

So, let's take a look: what would A times B be?2448

Well, we know what the A that we just figured out is: that is 3, 5, -2, 7;2450

and the B we started with, that we were given, is -2, 3, 0, 4.2456

So, we work this out; the 3, 5 on -2, 0 is going to get us a -6; 3, 5 on 3, 4 is going to get us 9 + 20, so 29.2463

-2, 7 on -2, 0 is going to get us -2 times -2...+4; and -2, 7 on 3, 4...-2 times 3 gets us -6; 7 times 4 gets us 28; add those together--you get 22.2477

And so, that is exactly the AB that we started with; so it checks out--our answer is good.2491

Great; all right, so that completes an understanding of determinants and inverses.2497

We have a great understanding of how that works right now.2500

We will see you at Educator.com later--goodbye!2503

However, if you want to check out the stuff for if we want to be able to do inverses for larger than 2 x 2,2505

larger than just that simple formula, let's take a look at it--let's look at that.2514

Finding the inverse of larger matrices: for the method we are about to discuss, we will need some techniques we haven't learned just yet.2519

In the first part of the next lesson, we discuss augmented matrices, row operations, and Gauss-Jordan elimination.2525

You will need to be familiar with these things before what we are about to talk about will totally make sense.2532

So, if you haven't already seen these things, go and check them out first, and then come back and watch this part.2537

It is just the first half of the next lesson--actually, probably more like the first third.2541

The method we are about to go over is applicable for finding the inverse of any m x n matrix.2546

If the matrix has no inverse, if it is singular, this method will end up failing.2550

It is normally easier to first check that there is going to be an inverse, before trying to put all of this work into it.2554

Just check to make sure that there is an inverse by getting the determinant,2559

because getting the determinant will actually take much less time that working through this method.2562

It is a good idea to check the determinant first to make sure that what you are doing will actually manage to work out.2566

All right, let's see how to do this: for an m x n matrix A, you begin by creating an augmented matrix with the identity matrix In.2571

So, if we have some A that is 1, 3, -2, -4, then we leave that part the same, and we drop in an identity matrix.2578

Since this is a 2 x 2, this ends up being a 2 x 2 identity matrix, right here.2586

We have 1, 3, 1, 0; -2, -4, 0, 1; we have it split in this middle, where the left side is A, and the right side is the identity matrix.2591

OK, the next step: you start applying the method of Gauss-Jordan elimination.2601

You use row operations to reduce A, that left side, to the identity matrix.2606

The result of the augmented matrix, once you manage to finally get this to be the identity matrix--2612

what you will have on the right side will be the inverse; you will have A-1 on the right side.2617

So, for example, if we have 1, 3, -2, -4, our first step is that we want to turn this into a 0.2622

We are doing Gauss-Jordan: so we take our row 2, and we add 2 times row 1; so 2 times 1 gets us +2;2629

that cancels to 0; 2 times 3 gets us +6 on -4; that goes to 2; 2 times 1 on 0 gets us 2;2636

2 times 0 on 1 is the same as it was before; so we have our new matrix here.2644

We continue with this method; we had 1, 3, 1, 0; 0, 2, 2, 1 on the previous slide.2648

So at this point, we want to turn this into a 1; so we multiply that entire row by 1/2.2653

This becomes 1; 2 times 1/2 becomes 1; 1 times 1/2 becomes 1/2.2659

At this point, we now want to get rid of this; we want to turn this into a 0 to continue with Gauss-Jordan elimination.2665

So, we subtract: we have a 1 here already, so we subtract 3 of row 2: so -3 times this:2670

1 times -3 on 3 gets us 0; and also, 0 times -3 gets us 1; we don't have any effect there.2678

Minus 3 here gets us -2; and -3 on 1/2 gets us -3/2.2686

So, at this point, we have an identity matrix here; this is just an identity matrix, because it is 1's on the main diagonal and 0's everywhere else.2692

So, what we have over here on the right side is our inverse matrix, A.2700

That is our inverse matrix A; we just bring it down and turn that into a matrix, and we have our answer.2706

Finally, you want to check your work; it is really easy to make a mistake in all of that arithmetic.2712

We were doing the simplest possible of 2 x 2; if you have to do this by hand, you are going to have to be at least doing 3 x 3 or larger.2716

So, it is really easy to end up making a mistake in all of that arithmetic.2722

So, make sure to do notations of what your row operations were, what we saw on the left there,2725

what we talk about in the next lesson when we explain this stuff.2729

And also, at the end, once you get to the very end, check your answer;2732

make sure that A-1 times A is equal to the identity matrix,2735

or A times A-1 is equal to the identity matrix.2738

So, for example, we started with A = this, and we figured out that A-1 should equal this.2741

So, we check our work: we multiply the two matrices together.2745

1, 3 times -2, 1: well, 1 times -2, plus 3 times 1...-2 + 3; that comes out to be 1.2750

1, 3 on -3, 2; that gets us -3, 2 + 3, 2, so that comes out to be 0; they cancel out.2758

Next, -2, -4 on -2, 1; -2 times -2 is 4, minus 4 times 2...4 - 4...that comes out to be 0.2766

And then, -2, -4 on -3, 2...3/2 times 1/2...so -2 times -3/2 gets us positive 6/2, minus 4/2; so we get 1 out of that.2775

Ultimately, it all checks out; we have figured out that this is, indeed, the inverse; it does end up working out just fine.2788

It is a really good idea to check your work at this point, because it is easy to make a mistake when you are doing that much arithmetic.2795

So, if you have to do this stuff by hand, always check your work at the end, because it is going to be a small amount of time2801

compared to the massive amount of time that you spend doing this.2805

And it would be a real shame if it ended up not being true.2807

All right, I hope that gives you a pretty good sense of how all this inverse and determinant stuff works.2810

And we will see you in the next lesson, when we finally get to see an application of just how powerful matrices are--2815

why we have been interested in them; it is because they allow us to do all sorts of really amazing things2820

that make things much easier than they would be from what we are used to so far--some pretty cool stuff.2825

All right, we will see you at Educator.com later--goodbye!2830

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