INSTRUCTORS Carleen Eaton Grant Fraser

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Dr. Carleen Eaton

Dr. Carleen Eaton

Matrix Multiplication

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Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
Section 2: Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
Section 4: Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
Section 5: Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
Section 7: Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
Section 8: Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
Section 9: Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
Section 10: Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
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
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
Section 11: Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (15)

0 answers

Post by Dejun Jing on July 4, 2020

FOr te=he matrix multiplcation you wrote 2(-2) is -2

2 answers

Last reply by: Dejun Jing
Sun Jul 5, 2020 3:10 PM

Post by Philippe Tremblay on July 28, 2014

At the very end, in your review, you state that you multiplied (B+C) by A. Beware as this is not the same as A(B+C). As you explained earlier, the commutative property does not apply to matrices.

2 answers

Last reply by: Dr Carleen Eaton
Tue Feb 19, 2013 8:25 PM

Post by Carroll Fields on February 17, 2013

ar approx. 28:30 into lecture, the calculation of example IV row 2 col 1, should be '0' not '1' as listed above. Making the final answer [5 13]
[0 -3]

similar problem as the one listed by Jeff Mitchell, please correct to prevent misinformation from being given.

1 answer

Last reply by: Carroll Fields
Sun Feb 17, 2013 12:50 PM

Post by julius mogyorossy on March 15, 2012

I like to imagine turning the first column 90 degrees to the right to see if they are multicompatible, then I keep turning the rows in the first column 90 degrees to the right, works for me.

5 answers

Last reply by: Dr Carleen Eaton
Thu Jul 31, 2014 7:13 PM

Post by Jeff Mitchell on March 6, 2011

at approx. 9:45 into lecture, the calculation for the value in row 2 column 1 --> the math was incorrectly done for 2(-2) which was simplified to (-2) and should have been -4. therefore, the sum value for Row 2,column 1 should be 4 and not 6.

Matrix Multiplication

  • Matrix multiplication is defined if and only if the number of columns of the first matrix is equal to the number of rows of the second matrix.
  • Matrix multiplication is associative and satisfies the left and right distributive properties.
  • Matrix multiplication is not commutative.

Matrix Multiplication

Find the product of matrix A and B
A = [
2
3
4
5
]B = [
1
2
4
5
]
  • Since the demensions of matrix A and matrix B match, you can proceed with multiplication.
  • One intuitive way to multiply matrices is to put the second matrix on top of the second one like this:

  • ⎡
    ⎢
    ⎢
    ⎣
    1
    2
    4
    5
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    2
    3
    4
    5
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    		⎤
    ⎥
    ⎥
    ⎦
  • The result from Rows and Columns will fall right into place on the resulting matrix.

  • ⎡
    ⎢
    ⎢
    ⎣
    1
    2
    4
    5
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    2
    3
    4
    5
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    R1C1
    R1C2
    R2C1
    R2C2
    		⎤
    ⎥
    ⎥
    ⎦
    		 =
    ⎡
    ⎢
    ⎢
    ⎣
    1
    2
    4
    5
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    2
    3
    4
    5
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    R1C1
    R1C2
    R2C1
    R2C2
    		⎤
    ⎥
    ⎥
    ⎦
    		 =
    ⎡
    ⎢
    ⎢
    ⎣
    1
    2
    4
    5
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    2
    3
    4
    5
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    (2)(1) + (3)(4)
    (2)(2) + (3)(5)
    (4)(1) + (5)(4)
    (4)(2) + (5)(5)
    		⎤
    ⎥
    ⎥
    ⎦
  • Simplify the resulting matrix
  • [
    (2)(1) + (3)(4)
    (2)(2) + (3)(5)
    (4)(1) + (5)(4)
    (4)(2) + (5)(5)
    ] = [
    2 + 12
    4 + 15
    4 + 20
    8 + 25
    ] = [
    14
    19
    24
    33
    ]

⎡
⎢
⎢
⎣
14
19
24
33
⎤
⎥
⎥
⎦
Find the product of matrix A and BA = [
− 1
2
3
− 2
]B = [
1
− 2
− 1
0
]
  • Since the demensions of matrix A and matrix B match, you can proceed with multiplication.
  • One intuitive way to multiply matrices is to put the second matrix on top of the second one like this:

  • ⎡
    ⎢
    ⎢
    ⎣
    1
    − 2
    − 1
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    		⎤
    ⎥
    ⎥
    ⎦
  • The result from Rows and Columns will fall right into place on the resulting matrix.
    ⎡
    ⎢
    ⎢
    ⎣
    1
    − 2
    − 1
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    R1C1
    R1C2
    R2C1
    R2C2
    		⎤
    ⎥
    ⎥
    ⎦
    		 =
    ⎡
    ⎢
    ⎢
    ⎣
    1
    − 2
    − 1
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    R1C1
    R1C2
    R2C1
    R2C2
    		⎤
    ⎥
    ⎥
    ⎦
    		 =
    ⎡
    ⎢
    ⎢
    ⎣
    1
    − 2
    − 1
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    ( − 1)(1) + (2)( − 1)
    ( − 1)( − 2) + (2)(0)
    (3)(1) + ( − 2)( − 1)
    (3)( − 2) + ( − 2)(0)
    		⎤
    ⎥
    ⎥
    ⎦
  • Simplify the resulting matrix

  • ⎡
    ⎢
    ⎢
    ⎣
    ( − 1)(1) + (2)( − 1)
    ( − 1)( − 2) + (2)(0)
    (3)(1) + ( − 2)( − 1)
    (3)( − 2) + ( − 2)(0)
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    − 1 + − 2
    2 + 0
    3 + 2
    − 6 + 0
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    − 3
    2
    5
    − 6
    		⎤
    ⎥
    ⎥
    ⎦

⎡
⎢
⎢
⎣
− 3
2
5
− 6
⎤
⎥
⎥
⎦
Find the product of matrix A and BA = [
− 1
2
3
− 2
]B = [
1
0
0
1
]
  • Since the demensions of matrix A and matrix B match, you can proceed with multiplication.
  • One intuitive way to multiply matrices is to put the second matrix on top of the second one like this:

  • ⎡
    ⎢
    ⎢
    ⎣
    1
    0
    0
    1
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    		⎤
    ⎥
    ⎥
    ⎦
  • The result from Rows and Columns will fall right into place on the resulting matrix.

  • ⎡
    ⎢
    ⎢
    ⎣
    1
    0
    0
    1
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    R1C1
    R1C2
    R2C1
    R2C2
    		⎤
    ⎥
    ⎥
    ⎦
    		 =
    ⎡
    ⎢
    ⎢
    ⎣
    1
    0
    0
    1
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    R1C1
    R1C2
    R2C1
    R2C2
    		⎤
    ⎥
    ⎥
    ⎦
    		 =
    ⎡
    ⎢
    ⎢
    ⎣
    1
    0
    0
    1
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    ( − 1)(1) + (2)(0)
    ( − 1)(0) + (2)(1)
    (3)(1) + ( − 2)(0)
    (3)(0) + ( − 2)(1)
    		⎤
    ⎥
    ⎥
    ⎦
  • Simplify the resulting matrix

  • ⎡
    ⎢
    ⎢
    ⎣
    ( − 1)(1) + (2)(0)
    ( − 1)(0) + (2)(1)
    (3)(1) + ( − 2)(0)
    (3)(0) + ( − 2)(1)
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    − 1 + 0
    0 + 2
    3 + 0
    0 − 2
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦

⎡
⎢
⎢
⎣
− 1
2
3
− 2
⎤
⎥
⎥
⎦
Find the product of matrix A and BA = [
1
− 2
− 1
0
]B = [
− 1
2
3
− 2
]
  • Since the demensions of matrix A and matrix B match, you can proceed with multiplication.
  • One intuitive way to multiply matrices is to put the second matrix on top of the second one like this:

  • ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    1
    − 2
    − 1
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    		⎤
    ⎥
    ⎥
    ⎦
  • The result from Rows and Columns will fall right into place on the resulting matrix.

  • ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    1
    − 2
    − 1
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    R1C1
    R1C2
    R2C1
    R2C2
    		⎤
    ⎥
    ⎥
    ⎦
    		 =
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    1
    − 2
    − 1
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    R1C1
    R1C2
    R2C1
    R2C2
    		⎤
    ⎥
    ⎥
    ⎦
    		 =
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    2
    3
    − 2
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    1
    − 2
    − 1
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    (1)( − 1) + ( − 2)(3)
    (1)(2) + ( − 2)( − 2)
    ( − 1)( − 1) + (0)(3)
    ( − 1)(2) + (0)( − 2)
    		⎤
    ⎥
    ⎥
    ⎦
  • Simplify the resulting matrix

  • ⎡
    ⎢
    ⎢
    ⎣
    (1)( − 1) + ( − 2)(3)
    (1)(2) + ( − 2)( − 2)
    ( − 1)( − 1) + (0)(3)
    ( − 1)(2) + (0)( − 2)
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    − 1 + − 6
    2 + 4
    1 + 0
    − 2 + 0
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    − 7
    6
    1
    − 2
    		⎤
    ⎥
    ⎥
    ⎦

⎡
⎢
⎢
⎣
− 7
6
1
− 2
⎤
⎥
⎥
⎦
In problem number 4 you multiplied [
1
− 2
− 1
0
]*[
− 1
2
3
− 2
] = [
− 7
6
1
− 2
] and in problem 2 you multiplied [
− 1
2
3
− 2
]*[
1
− 2
− 1
0
] = [
− 3
2
5
− 6
]
  • What do you notice about matrix property of multiplication?
I noticed that matrix multiplication is not commutative. In other words A*B ≠ B*A
Find [
− 2
3
0
0
3
− 1
]*[
− 1
2
1
4
− 2
0
]
  • Since the dimensions match 2x3 and 3x2, 3 = 3. You can proceed with the multiplication.
  • Use the same strategy as with previous problems.

  • ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    − 1
    2
    1
    4
    − 2
    0
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 2
    3
    0
    0
    3
    − 1
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    		⎤
    ⎥
    ⎥
    ⎦

  • ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    − 1
    2
    1
    4
    − 2
    0
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 2
    3
    0
    0
    3
    − 1
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 2( − 1) + 3(1) + 0( − 2)
    − 2(2) + 3(4) + 0(0)
    0( − 1) + 3(1) + − 1( − 2)
    0(2) + 3(4) + − 1(0)
    		⎤
    ⎥
    ⎥
    ⎦
  • Simplify the resulting matrix

  • ⎡
    ⎢
    ⎢
    ⎣
    − 2( − 1) + 3(1) + 0( − 2)
    − 2(2) + 3(4) + 0(0)
    0( − 1) + 3(1) + − 1( − 2)
    0(2) + 3(4) + − 1(0)
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    2 + 3 + 0
    − 4 + 12 + 0
    0 + 3 + 2
    0 + 12 + 0
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    5
    8
    5
    12
    		⎤
    ⎥
    ⎥
    ⎦

⎡
⎢
⎢
⎣
5
8
5
12
⎤
⎥
⎥
⎦
Find [
− 1
1
− 2
2
4
0
]*[
− 1
2
1
4
− 2
0
]
  • Since the dimensions match 2x3 and 3x2, 3 = 3. You can proceed with the multiplication.
  • Use the same strategy as with previous problems.

  • ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    − 1
    2
    1
    4
    − 2
    0
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    1
    − 2
    2
    4
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    		⎤
    ⎥
    ⎥
    ⎦

  • ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    − 1
    2
    1
    4
    − 2
    0
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1
    1
    − 2
    2
    4
    0
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    − 1( − 1) + 1(1) + − 2( − 2)
    − 1(2) + 1(4) + − 2(0)
    2( − 1) + 4(1) + 0( − 2)
    2(2) + 4(4) + 0(0)
    		⎤
    ⎥
    ⎥
    ⎦
  • Simplify the resulting matrix

  • ⎡
    ⎢
    ⎢
    ⎣
    − 1( − 1) + 1(1) + − 2( − 2)
    − 1(2) + 1(4) + − 2(0)
    2( − 1) + 4(1) + 0( − 2)
    2(2) + 4(4) + 0(0)
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    1 + 1 + 4
    − 2 + 4 + 0
    − 2 + 4 + 0
    4 + 16 + 0
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    6
    2
    2
    20
    		⎤
    ⎥
    ⎥
    ⎦

⎡
⎢
⎢
⎣
6
2
2
20
⎤
⎥
⎥
⎦
Find [
1
0
− 1
1
0
− 1
]*[
− 1
1
0
0
2
− 2
]
  • Since the dimensions match 2x3 and 3x2, 3 = 3. You can proceed with the multiplication.
  • Use the same strategy as with previous problems.

  • ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    − 1
    1
    0
    0
    2
    − 2
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    1
    0
    − 1
    1
    0
    − 1
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    		⎤
    ⎥
    ⎥
    ⎦

  • ⎡
    ⎢
    ⎢
    ⎢
    ⎣
    − 1
    1
    0
    0
    2
    − 2
    		⎤
    ⎥
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    1
    0
    − 1
    1
    0
    − 1
    		⎤
    ⎥
    ⎥
    ⎦
    ⎡
    ⎢
    ⎢
    ⎣
    1( − 1) + 0(0) + − 1(2)
    1(1) + 0(0) + − 1( − 2)
    1( − 1) + 0(0) + − 1(2)
    1(1) + 0(0) + − 1( − 2)
    		⎤
    ⎥
    ⎥
    ⎦
  • Simplify the resulting matrix

  • ⎡
    ⎢
    ⎢
    ⎣
    1( − 1) + 0(0) + − 1(2)
    1(1) + 0(0) + − 1( − 2)
    1( − 1) + 0(0) + − 1(2)
    1(1) + 0(0) + − 1( − 2)
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    − 1 + 0 + − 2
    1 + 0 + 2
    − 1 + 0 + − 2
    1 + 0 + 2
    		⎤
    ⎥
    ⎥
    ⎦    		 =⎡
    ⎢
    ⎢
    ⎣
    − 3
    3
    − 3
    3
    		⎤
    ⎥
    ⎥
    ⎦

⎡
⎢
⎢
⎣
− 3
3
− 3
3
⎤
⎥
⎥
⎦

*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

Matrix Multiplication

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
  • Dimension Requirement 0:17
    • n = p
    • Resulting Product Matrix (m x q)
    • Example: Multiplication
  • Matrix Multiplication 3:38
    • Example: Matrix Multiplication
  • Properties of Matrix Multiplication 10:46
    • Associative Property
    • Associative Property (Scalar)
    • Distributive Property
    • Distributive Property (Scalar)
  • Example 1: Possible Matrices 13:31
  • Example 2: Multiplying Matrices 17:08
  • Example 3: Multiplying Matrices 20:41
  • Example 4: Matrix Properties 24:41

Transcription: Matrix Multiplication

Welcome to Educator.com.0000

In today's lesson, we will be covering matrix multiplication.0002

In a previous lesson, we discussed scalar multiplication, which is multiplying a constant by a matrix.0005

This time, we will be talking about multiplying one matrix by another.0012

Before you proceed with matrix multiplication, you need to verify that the dimension requirement has been met.0018

So, suppose that matrix A is m by n, and matrix B has dimensions of p by q.0024

The product of these two matrices can be obtained only if n = p.0031

Now, what is that saying? What that is saying is that the number of columns of the first matrix must equal the number of rows of the second matrix.0036

So, in order to find AB, to find that product, to be allowed to do that kind of multiplication,0045

the number of columns of the first matrix must equal the number of rows of the second matrix.0052

OK, once you find that product, the resulting product matrix, AB, will have dimensions of m times q.0099

So, the product matrix dimensions will have the number of rows of the first matrix and the same number of columns as the second matrix.0109

To make this more concrete, let's look at an example.0115

If matrix A equals 2, -1, 4, 0, and matrix B equals 3, 4, 2, 1, 0, -3, 0, -2; my first question is, "Can I multiply them--can I find AB?"0118

Well, this has one row and four columns; it is a 1x4 matrix.0137

This has four rows and two columns; it is a 4x2 matrix; so this gives me 1x4 and 4x2.0145

So, all I have to do is see, "OK, does this second number equal this first number?" and yes, the second number does equal the first number.0152

Therefore, I can multiply those.0162

Now, what are going to be the dimensions of the product matrix?0164

Well, the dimensions of the product matrix...this is a 1x4, multiplied by a 4x2 matrix; and AB is going to have0168

the same number of rows as this first matrix (which is 1), and the same number of columns as the second matrix (which is 2).0175

So, the product matrix is going to be a 1x2 matrix.0184

So, the important thing is: before you multiply, make sure that you verify that the dimension requirement is met--0188

that the second number of the first matrix is equal to the first number of the second matrix.0193

And to predict the dimensions of the product matrix, you take this first number and this second number; and that will give you the product matrix dimensions, 1x2.0200

OK, multiplying by matrices is not exactly what you would expect.0217

It is not like addition and subtraction of matrices, where (in addition and subtraction) we just took the first matrix0224

and added the corresponding element of that to the corresponding element of the second matrix (and the same in subtraction).0231

You might think, "OK, I am just going to multiply the corresponding elements by each other."0238

But it is actually more complicated than that; and you need to take it step-by-step.0243

So, the best way to understand this is to go through an example: so let's look at a pair of matrices and multiply them out.0247

OK, this is my first matrix; I am going to call it A; and then, I want to multiply it by another matrix, which I am going to call B.0264

So, before I multiply, I have to make sure that they meet the dimension requirements.0277

And A has two rows, and it has four columns; B has four rows and two columns.0284

So, I am allowed to multiply these, because this second number equals the first number.0292

My product matrix, AB, is going to have two rows and two columns; it is going to be a square matrix with dimensions 2x2.0299

Now, as you read up here, it says that the element in row I and column J of the product...0311

so the element in a certain row and column of the product of the matrices A and B...0317

is obtained by forming the sum of the products of the corresponding elements in row I0323

(in a certain row of matrix A) and the corresponding column, J, of matrix B.0329

What does this mean? Well, instead of just saying row I and column 1, let's start with row 1, column 1.0335

I want to find the element that goes right here, in row 1, column 1.0345

And the way I am going to do that is: I am going to go over here to row 1, and I am going to go here in column 1;0350

and I am going to multiply 3 by 4 and find that product, and then I am going to add that to the next product, 2 by 0.0361

Then, I am going to add that to the next product, 0 by 1, to 0 by 3, and so on.0369

So, row 1, column 1; then I am going to go to the corresponding row in A and the corresponding column in B.0375

And working this out, this gives me 3 times 4, plus 2 times 0, plus 0 times 3, plus 1 times -2.0380

Working that out, that is going to give me 12 + 0 + 0 - 2; 12 - 2 is 10.0399

Therefore, my row 1, column 1 element is 10.0410

OK, next let's work on row 1, column 2; that means I am going to go to row 1 here in A, and column 2 here, in B, and do the same operations.0414

So, row 1, column 2: that is 3 times -1, plus 2 times 2, plus 0 times 1, plus 1 times 0.0429

OK, working that out, this gives me -3 + 4 + 0 + 0; that is 4 and -3, so that is 1.0446

Therefore, my row 1, column 2 element for the product is 1.0461

All right, now I need to work with this second row; and I am going to think about what position this is right here.0468

This is row 2, column 1; that means I am going to go to row 2 here and column 1 in B and do the same thing.0472

OK, row 2, column 1: -1 times 4, plus 0 times 0, plus 4 times 3 (so the third element here and the third element here),0487

plus the fourth element here and the fourth element there.0507

All right, figuring this out, this is going to give me -4 + 0 + 12 + -2, or - 2; look at it either way.0512

This is going to give me -4 and 12, which is 8, minus 2...actually, let's make this clearer...plus 12, minus 2...it is going to give me...0526

let's see...that is -4, 0, 4 and 3, 2 and -2; OK, so this is going to give me 12 - 6, or 6.0546

Now, the next row and column position: I have row 2, column 2.0562

Here is row 2; here is column 2; it is going to give me -1 times -1, plus 0 times 2, plus 4 times 1, plus 2 times 0.0577

OK, so that is going to come out to...-1 times -1 is 1, plus 0, plus 4, plus 0; 4 + 1 is 5, so I am going to get 5 right here.0601

So again, if I just looked here, and I said, "OK, that is row 1, column 2," then I would get that0618

by going to row 1 here and column 2 here and multiplying each of those, and then finding the sum of their products.0626

So, even with a bigger matrix, you can always find a particular position by using this method.0638

OK, there are some properties that govern matrix multiplication that you need to be familiar with.0646

If A, B, and C are matrices for which products are defined, and k is any scalar, then these properties hold.0653

The first property you will recognize as the associative property for multiplication.0660

And what it says is that I can either multiply A times B, find that product, and then multiply by C;0667

or I can multiply B times C, find that product, and then multiply A; and I will get the same result.0679

OK, next I see the same thing here (it is the associative property), except this time, it also involves multiplying a scalar.0687

So, here I have matrix multiplication, and then I also have a scalar.0695

And I can either multiply the two matrices and then multiply the scalar times that product;0698

or I can multiply the scalar times the first matrix, then multiply that product by the second matrix;0704

or the scalar and the second matrix, and the product by the first matrix.0712

So, it doesn't matter which order I do the multiplication in, in this situation; these two first, then the other two.0716

Next, you are going to recognize the distributive property.0726

And the distributive property, as usual, says that, if I have these two matrices, B + C,0732

and I am multiplying A by them, another approach that would give the equal result0739

is to first multiply A and B, and then add that product to the product of A and C--the distributive property.0743

And again, the distributive property would work if I placed these as follows, added A and B, and then multiplied C.0751

The same thing: I could say the product of A and C, plus the product of B and C.0763

What is very important to realize is that the order that you multiply matrices in matters a lot.0768

So, if I say, "Oh, I am going to do AB; does this equal BA?" no, it does not always equal that.0775

Sometimes it does; but very often it does not--these are two different things.0783

Therefore, matrix multiplication is not commutative; it does not follow the commutative property,0788

and so you can't simply say, "A times B is the same as B times A"; you can't just change the order of those two.0797

The first example: suppose A is 3x4; B is a matrix that is 4x3; C is 3x3; and D is 4x4.0812

What are possible defined products of the four matrices?0821

Recall that, for a product to be defined, the second number of the first matrix has to equal the first number of the other matrix.0824

So, let's first look at AA; is that defined?0834

Well, if I have 3x4, and I am trying to multiply it by 3x4; these two are not equal, so this is not defined.0840

OK, and let's make a list here of the ones that are defined, because that is what they are asking me for--where the product is defined.0852

If I take A times B, well, the second number of A is equal to the first number of B; so that is defined.0863

Now, let's do it in the opposite order, BA: if I have 4x3 and 3x4, that is also defined.0871

OK, now AC: 3x4 and 3x3--that is not defined.0882

Now, if I put C first--if I say CA--it is 3x3 and 3x4, and these two are equal, so CA would have a defined product.0888

OK, now 3x4 and 4x4; that is AD; that is defined, because of the 4 and the 4.0903

However, if I put D first--if I say I am going to do DA--that is going to give me the D first, which is 4x4, and then A, 3x4; that is not defined.0910

OK, moving on to B: B times B (BB) is 4x3 and 4x3; that is not defined.0923

BC is 4x3 and 3x3; these two are equal, so BC is defined.0934

CB is 3x3 and 4x3--not defined; OK, BD--4x3 and 4x4--not defined.0940

But if I put the D first, then I would get 4x4 and 4x3; that is defined, so DB is defined.0953

OK, now I am up to C; 3x3 and 3x3--CC--that is a square matrix, so that is defined; the multiplication is defined for that.0965

OK, now, CD--3x3 and 4x4--is not defined; then I try the D first--it is not going to matter: 4x4 and 3x3 is still not defined.0975

OK, let's go to D: I can multiply DD, because I have 4x4 and 4x4, and that means that, of course,0988

since this is a square matrix, this second number and the first number there are going to be equal.1000

So, these are my defined products; I have 3, 6, 8 defined products.1005

So, these are all the ones where the second number of the first matrix--1010

the number of columns--was equal to the first number of the second matrix--the number of rows.1016

And I can perform multiplication of these sets of matrices.1021

OK, now doing some matrix multiplication: before I proceed, I am going to make sure that the product is defined--that I can do it.1029

So, I have a 2x2 matrix here, and I have a 2x2 matrix here; therefore, multiplication is allowed.1038

I am rewriting this down here, since this second number is equal to this first number (so multiplication is allowed).1052

Using the method we discussed: my product matrix is going to be 2x2 also--the first number here and the second number there.1066

I am expecting a 2x2 matrix; so let's first look for this position--row 1, column 1.1076

I am going to go to row 1 here and column 1 here and work with those two.1086

I am going to say 3 times 0 (that product), and I am going to add it to the product of 2 times 4.1091

That is going to give me 0 + 8, which is 8; so the element in row 1, column 1 is 8.1099

Now, row 1, column 2--I am going to go to row 1 here--row 1 of this first matrix--and column 2 of the second matrix.1111

And that is going to give me...row 1 is 3, times 6; so row 1, column 2 is 2 and -1, so plus 2 times -1.1124

It is going to give me 18 - 2, which equals 16.1140

OK, now the next position is row 2, column 1; working with that, that is going to give me a -1 times 0, plus 4 times 4,1150

which is going to equal 0 plus 16, which is 16.1170

Next, I want to find...this is row 2, but it is column 2 this time.1177

So, I go to row 2 here and column 2 here: -1 times 6, plus 4 times -1--that is going to give me -6 - 4, or -10.1185

As expected, my product matrix is also 2x2; and again, if I picked any element in here--1208

let's say I picked this 16--I could simply say, "OK, that is row 2, column 1."1215

And I would get that by multiplying and finding the products of row 2 of the first matrix1223

and column 1 of the second matrix, and adding those up--finding the sum of those.1230

This is the result of the multiplication of these two matrices.1236

Example 3: I have 1, 2 rows and 3 columns, so I have a 2x3 matrix and a 3x2 matrix.1244

These two are equal, so I am allowed to multiply them.1255

The product of these two is going to have this number of rows (2) and this number of columns.1258

So, I am going to get a matrix that is going to be 2x2 for my product here.1267

OK, I am setting up the product matrix right here, and first looking for row1 , column 1; so I am going to use row 1 here, column 1 here.1275

2 times 0 is the product; then I am going to add that to the next product, which is -1 times 3, and the third product, which is 0 times -2,1293

which equals...2 times 0 gives me 0, minus 3, plus 0; so it is -3--row 1, column 1, right here, is -3.1314

Row 1 right here, but column 2: I am going to take 2 times 1; OK, in row 1, column 2, the next set of elements is -1 and 6,1331

so plus -1 times 6; then I still have row 1, column 2; I have 0 times -1; figure that out--1355

that is 2 - 6 + 0, so it is just 2 - 6, and that gives me -4, so row 1, column 2 is -4.1368

OK, moving on to the second row: the first thing I need to find is row 2, column 1--row 2 here, column 1 here.1380

0 times 0, plus 3 times 3, plus -2 times -2; OK, that is going to give me 0 + 9, and then -2 times -2 is positive 4, so plus 4.1392

9 + 4 is 13, so row 2, column 1, gives me 13.1412

OK, next I need to find row 2, column 2.1417

Row 2, column 2: 0 times 1, plus 3 times 6, plus -2 times -1 is going to give me 0 + 18... -2 and -1 is positive 2, so that is 20.1425

OK, and as expected, I got a 2x2 matrix, and I could find any position by just honing in and saying,1448

"OK, that is row 1, column 2, so I am going to multiply row 1 elements and column 2 elements and add those products."1454

Now, at first, I know this seems like a lot of work; but you should go step-by-step, find the row, find the column,1462

write out what you need to write out; and then, later on, as you get faster, you can just do a lot of it in your head.1467

But right now, since there are a lot of steps, go to each row and each column, and figure out what those are.1473

OK, Example 4 asks us to do some multiplication and some addition, so I need to find the sum of the products AB + AC.1479

But remembering the distributive property, I can actually do this more easily.1494

Recall that, according to the distributive property for matrix multiplication, AB + AC = A(B + C).1499

I would rather do it this way, because matrix multiplication is difficult; addition is much easier.1516

This way, instead of multiplying twice and adding once, I would rather just add these together, and then only have to multiply one time.1522

So, I am going to approach this by using the distributive property.1532

So, what I am going to do is find A times (B + C), and it is going to be equivalent to this.1536

Let's start out by finding B + C: B is 0, 3, 1, 4; that is B; and I am going to add that to C, which is right here: 1, 2, 0, -3.1543

Now, recall from matrix addition: all you have to do is add the corresponding elements,1567

and you are going to get a matrix of the same dimensions as the original.1572

And these are both 2x2 matrices, so I can add them.1575

0 + 1 is going to give me 1; 3 + 2 is going to give me 5; 1 + 0--I am going to get 1; 4 - 3 is 1.1578

Now, this gives me B + C; the next thing I need to do is multiply A times B + C.1590

So, let's go back up here and look at what A is: A is...2 and -1 are the elements, and 3, 2; that is A.1602

I am going to go ahead and add that to what I discovered that B + C is: 1, 5, 1...oh, excuse me, multiply--we are now multiplying that.1611

All right, recall from matrix multiplication that I have to make sure I am even allowed to multiply these.1624

And this is a 2x2 matrix, and this is a 2x2 matrix; so this second number is equal to the first number here, so I can multiply them.1635

All right, row 1, column 1--this position on my product matrix--is going to give me row 1 here and column 1 here.1644

That is 2 times 1, plus 3 times 1; that is just 2 + 3, so that is 5.1653

OK, now, row 1, column 2: row 1 here, and column 2 here: 2 times 5, plus 3 times 1.1664

2 times 5 is 10, plus 3--that is going to give me 13 for this position.1678

OK, the next row, row 2, column 1: -1 times 1, plus 2 times 1; that is going to give me -1 plus 2, which equals 1.1685

So, for row 2, column 1, I get 1.1707

Now, row 2, column 2, right here, is -1 times 5, plus 2 times 1.1710

-1 times 5 is -5, plus 2 gives me -3.1725

So, this gives me A times (B + C), which is equivalent to what they asked me to find, AB + AC.1732

So, the key step here was recognizing that you could use the distributive property there, because that made a lot less work.1739

I only had to do one set of matrix multiplication, instead of 2.1745

So, I first used the distributive property; and then I had to add B and C.1750

So, I added B and C, and got this sum matrix, B + C.1757

And then, I multiplied it by A; so here is (B + C) times A, using my typical method to get this result.1761

That concludes this lesson of Educator.com on matrix multiplication, and I will see you next time!1770

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