In the first section of this chapter we saw how to multiply a number by a matrix.  Now we will consider the product of two matrices. The motivation for defining matrix products comes from studying systems of equations. In Section 6 of this chapter we will be ready to use matrices to solve a system of equations.

Let's begin by considering the following equation.

We will write the coefficients on the left side in a 1 × 3 matrix (a row matrix) and the variables in a 3 × 1 matrix (a column matrix). The 8 on the right side of the equation is written in a 1 × 1 matrix, shown below.

We can return to our original equation by multiplying the members of the row matrix by those of the column matrix, and then adding.

We can begin to define matrix multiplication by looking at this special case.  We have a row matrix A  and a column matrix B, whose product AB is a 1 × 1 matrix, having the single member 8 (also called 4x - 2y + z ).

In general, the product of two matrices is found by multiplying rows and columns. However, we need to consider proper notation, some limitations on multiplication, and then the procedure.  

Notation:

When talking about matrices we often refer to each element's location in the matrix as the ith row and the jth column. You may see this notation in textbooks written as

 

This is read "A sub i j".  You should think of this as "the element of matrix A in the ith row and jth column.  It is also common notation to see matrices labeled in a general fashion as shown below.

Limitations:

The product of matrices A and B can be computed if matrix A has n columns and matrix B has n rows, regardless of the other dimensions.  The product will have as many rows as A and as many columns as B.

This means that you can multiply two matrices only if the number of columns in the first matrix is equal to the number of rows in the second matrix.  Some examples follow:

 

2 × 2	2 × 3		2 × 3	2 × 2
possible			not possible

Procedure:

To multiply matrix A and B, multiply every element in the ith row of A and the jth column of B. Then add the results to find the element in the ith row and jth column of AB.

example: Find  AB
Notice the results of multiplication are: