In our unit on Determinants, we briefly discussed the question, "What is a matrix?" On this page we'll cover some of the same material, in case you haven't yet done that unit. If you already know this material, you can skip down to the questions at the bottom!

First, you should be aware that you will see both the word "matrix" and the word "matrices." Matrix is the singular form of the word and matrices is the plural. Please don't say "matrixes" - that's as strange as saying "mouses" or "gooses."

A matrix is nothing more than an array (rows and columns) of numbers, written with square brackets (or large parentheses) around them. The following is an example of a matrix:

Note that the number of rows and columns do not have to match. In fact, we could have a matrix with just one row, or a matrix with just one column:

In Algebra, you have probably grown accustomed to using lower case letters as variables representing unknown quantities. And although using lower case letters is not a hard-and-fast rule, it's a good habit, because we're typically going to use upper case letters as variables to represent matrices.  If you follow this habit, it will make life simpler, because then if you see this: kA, you'll know that it's a number times a matrix, and if you see this: AB, you'll know it's two matrices being multiplied together.

It is important to remember that a matrix is just a collection of numbers in columns and rows; the matrix itself does not have a value. In the unit on Determinants you will learn that all square matrices (matrices that have the same number or rows and columns) have a determinant, and that determinant has a value, but the matrix itself is just a block of numbers!

Every matrix has dimensions, and the dimensions are a way of describing how big the matrix is. If we say that a matrix is a 2 x 3 matrix, that means it has 2 rows, and 3 columns.  The matrix below is a 4 x 2 matrix, because it has 4 rows and two columns.

Another way of saying it is, the first number in the dimensions represents how tall the matrix is, and the second number represenets how wide it is.

Why is it important to understand the dimensions of a matrix? Well, one big reason is that matrix operations (adding, subtracting, etc.) can only be done on matrices that have specific dimensional requirements. For example, you can only add two matrices if they have the same number of rows and the same number of columns.

We could refer to any entry of the matrix using a variable and a subscript. For example, a_1,2 is the element in the first row, second column, and a_3,1 is the element in the third row, first column. In the matrix above, a_1,2 = 7, and a_3,1 = 3.