Algebra 2 Basic Matrix Concepts

Welcome to Educator.com.0000

Today's lesson introduces the concept of matrices.0002

And matrices are used throughout math and science as an approach to problem solving.0006

In this course, we are going to use them to solve systems of equations.0013

However, they are also used in fields such as physics, computers, and genetics.0018

First of all, defining what a matrix is: a matrix is a rectangular array containing variables or constants, which is enclosed by brackets.0026

And the plural form of the word "matrix," which I just used, is matrices; so you will hear me say that in the course.0037

And to give you an example of what a matrix looks like, as I said, a matrix is enclosed in brackets, and it contains variables or constants.0045

And starting out, we are going to be working with matrices that contain constants.0062

And then, towards the end of this series of lectures, we will see a matrix that involves variables.0066

This is an example of a matrix; and these are usually designated by capital letters, so I could call this matrix A.0079

Another matrix, B, might be smaller, perhaps containing only two numbers.0087

Another matrix, C, could have a different set of numbers.0097

Each variable or constant within a matrix is called an element; so, 2 is an element; 1 is an element; each of these is an element.0108

And later on, we will also see matrix equations, because you can perform operations with matrices.0118

For example, I could have a matrix equation that says A + B = C.0126

So, I could have some matrix, A, that when added to B equals another matrix C.0134

So, just as we performed operations on constants and terms, we can perform operations on matrices.0138

OK, starting out, it is important to understand the concept of dimensions when referring to a matrix.0148

And with a matrix, dimensions refers to the number of rows and columns in the matrix.0154

For example, let's say I have a matrix, and you can see, of course, that it has a certain number of rows and a certain number of columns.0161

Well, m refers to the number of rows; and here, that is 1, 2, 3, 4--I have 4 rows.0179

For this matrix, looking at columns (and we will say n here is the number of columns), I have 1, 2 columns.0191

Therefore, I would call the dimensions of this matrix 4 by 2; this is a 4x2 matrix.0200

And this is important, because the dimensions often limit what you are able to do--0207

which operations you can perform on a set of matrices.0214

So, looking at another matrix: this matrix has 1, 2 rows and 1, 2, 3, 4 columns; so this is a 2x4 matrix.0219

It is important--the order of the numbers is very important: here it is 4x2 (4 rows, 2 columns); here it is 2x4, because it's 2 rows, 4 columns.0243

OK, there are certain matrices that are special cases: a 1xn matrix has one row, and is called a row matrix.0262

For example, this is a row matrix, because it has one row (in this first term here, it tells the number of rows).0272

And looking at the number of columns, I have 1, 2, 3, 4, 5, 6; so this is a 1x6 matrix, and it is a row matrix, because it has only one row.0286

A second type of special matrix is a column matrix; here, it is an mx1 matrix--it has one column.0300

So here, it is 1 row; and here, 1 column.0309

For example (let me make this a little bit more spread out)...2, 3, 8, 4, 2: OK, so here, I have 1, 2, 3, 4, 5 rows, but I have only 1 column.0312

So, this would be a 5x1 matrix, and since it has a 1 right here, this is a column matrix.0346

A third type of special matrix is called a zero matrix; and this zero matrix has all its elements equal to 0.0358

For example, this would be an example of a zero matrix.0365

So, these are three special types of matrices: a row matrix with one row, a column matrix that has one column,0375

and a zero matrix, which contains elements that are all 0.0382

We say that two matrices are equal if they have the same dimensions, and their corresponding elements are equal.0389

Let's talk about what corresponding elements are, using this set of matrices as an example.0396

OK, so I am going to call this A, and this matrix B; and I can say that A = B, because the corresponding elements of A and B are equal.0415

Elements are corresponding if they have the same position within a matrix.0424

By "position," I mean the same row number and the same column number.0428

So, right here, this element is in row 2, column 1; its corresponding element is also going to be in row 2, column 1.0432

And these are the same; and my row 1, column 1 elements are the same; and so on.0450

For each position, all of the elements are the same--the corresponding elements; therefore, these two matrices are equal.0456

If I had the same numbers, but they weren't in the same positions, the two matrices would not be equal.0463

The same dimensions (and I do have the same dimensions--1, 2, 3 rows, 2 columns; this is a 3x2 matrix,0471

and the same here--1, 2, 3 by 2 matrix) and corresponding elements are all equal: 1 equals 1, 3 equals 3, and so on.0477

I can say that these two matrices are equal.0487

OK, doing some examples: first, we are asked, "What are the dimensions of this matrix?"0493

And remember that when we do dimensions, we say mxn, so we look first at the number of rows, times the number of columns.0500

So, all I need to do is count the number of rows (that is 1, 2, so m = 2, and that is the number of rows),0513

and columns (n = 1, 2, 3--there are three columns); therefore, this is a 2x3 matrix; the dimensions of this matrix are 2x3, rows by columns.0523

Example 2 (a different matrix here): What are the dimensions of this matrix?0544

Again, we are looking at rows by columns; I have 1, 2 rows; 1, 2, 3, 4, 5 columns.0549

So, this is a 2x5 matrix, because it contains 2 rows and 5 columns; always have rows first, then columns.0568

Write the 3x3 zero matrix: this is telling me that I am going to have 3 rows and 3 columns.0579

And since it is a zero matrix, all elements are zero; OK, so that is going to give me 1, 2, 3 columns and 1, 2, 3 rows.0588

I just need to fill in, and all elements are 0; and again, a matrix is contained within brackets.0606

This is a 3x3 matrix (3 rows, 3 columns); it is a zero matrix; and I have my brackets around it to indicate that it is a matrix.0613

This is a 3x3 zero matrix.0622

Example 4: Write a matrix that is both a row matrix and a column matrix.0628

Recall that a row matrix has one row, and then some number of columns.0632

A column matrix has some number of rows, but only one column.0643

Putting this together, this is telling me that I am going to have...0649

if it is both a row and a column matrix, that means that in this m position,0653

the rows are going to be 1; and the columns are also going to be 1.0659

This is a 1x1 matrix that they are asking for; and I could use any constant or variable.0662

This would be both a row matrix and a column matrix; it has one row, and it has one column.0670

Or it could be 5, or I could use a variable (such as y or x).0675

So, any of these examples would be a row matrix and a column matrix at the same time.0680

That concludes this lesson from Educator.com; and I will see you next time, when we talk more about matrices.0689