Matrices Defined

Definition

A matrix is just another name for a grid of numbers.  Like any grid, it has rows located along the horizontals and it has columns located along the verticals.  A matrix with 2 rows and 3 columns is called a 2x3 matrix.  In general, a matrix with N rows and M columns is called an MxN matrix.  Here is an example of a 2x3 matrix, we give it the variable name A.

Matrix Entries

If we want to refer to a particular number in the matrix, called an entry (or sometimes, an element), we can identify it using a pair of indices. The entry found at the j^th row and the k^th column is written in the form a_j,k.  So for the matrix above, if we use a_j,k to refer to its entries, then a_2,3 is 8.  Here is a picture to make things clearer.

Alternate Notation

Using the notation for the entries of a matrix, sometimes matrices are written in terms of those entries.  If A is a matrix having n rows and m columns with entries a_i,j, then we can write:

This is just another way to write the matrix A.  Often n and m can be inferred from the context, and in that case you may just see something like A = (a_i,j).

Matrix Shape

Several arithmetic properties of matrices depend upon the shape of the matrices involved.  Most often when we need are interested in the shape of a matrix, we are really only care whether or not it has the same number of columns as it has rows.  When a matrix does have the same number of columns as it has rows then the matrix is said to be square.  Here are examples of square and non-square matrices.

Adding Matrices

Matrices can be added and subtracted with one another.  This kind of arithmetic works almost exactly like you would expect - when adding two matrices together, you just add their components.  Here is an example:

Knowing When To Add

As you can see from the previous example, we added two matrices by adding up an entry from the same position in each matrix.  It then makes sense that two matrices cannot be added together unless they have the same shape.  If A and B are two matrices, then the following rule determines whether or not they can be added:

In the first case, A and B can be added because both have the same shape - they each have n rows and m columns.  In the second case, A and B cannot be added because they have different numbers of rows and columns.  (Of course, if m = r and n = s, then this is just the first case again, but we assume that m and n are different from r and s.)

The Zero Matrix and Additive Inverses

Look at this example:

Adding these two matrices yielded a matrix with a zero in every entry.  Such a matrix is called, sensibly enough, a zero matrix, and is often denoted by 0, or [0].  If A and B are two matrices, and if it is legal for us to add them, then A is called the additive inverse of B if and only if A+B = 0.

Multiplication By Scalar Values

We just saw how we can add two matrices together.  Now we will see another arithmetic operation involving matrices called scalar multiplication. Given a matrix, we can multiply that matrix by a number, called a scalar. Scalar multiplication involves multiplying each entry in a matrix by the scalar value.   Here is an example:

You can see that we just multiplied each entry by the scalar value 2.  A general way to write this, using the notation introduced above, is like this:

So given a matrix A, and a scalar value k, then the matrix kA is defined by multiplying each entry a_i,j of A by k.

Subtraction

Using what we know about scalar multiplication, we can to define matrix subtraction.  If A and B are two matrices and we want to subtract B from A, then we just add -1 times B to A.   That is:

A - B = A + -1B

In that case, another way to write the additive inverse of A is -A, since A + (-A) = A-A = 0.

Algebraic Properties

Let A, B and, C be matrices each having n rows and m columns, and let k be a scalar value.  The the following properties hold:

• associative:      (A + B) +C = A + (B + C)
• commutative:   A + B = B + A
• distributive:      k(A + B) = kA + kB