Scalar multiplication with matrices is not too difficult once you get in the routine of performing the process. You can practice more problems like those above on the accompanying worksheet.

A second type of multiplication is to multiply two matrices together and it is a little more involved.

Multiplication of matrices has different rules than addition and subtraction. For

matrix multiplication, the columns of the first

matrix MUST match with the rows of the second matrix.

A = and B = .

To multiply AB, we first have to make sure that the number of columns in A is the same as the number of rows in B.

Matrix A has 2 columns and

matrix B has 2 rows so we will be able to perform this operation. The dimension of the new

matrix will be defined as:

Rows = number of rows in A

Columns = number of columns in B

An easier way to look at these dimensions is shown in the following figure:

Let’s do the multiplication and call our new

matrix M (AB = M). This

matrix will be a

. We simply need to fill in the six entries in this matrix.

Remember we are working with the matrices A and B shown below.

A = and B = .

We begin by multiplying (4)(2) and adding that to (6)(–2). That is now the

entry in m

_{11}.

Now multiply (4)( –3) and add that to (6)(0). This is the

entry in m

_{12}.

Multiply (4)(1) and add that to (6)(5). This is the

entry in m

_{13}.

We now have the top row of our answer

matrix M.

We’ll follow the same process to fill in the second row of the answer matrix.

Entry for m_{21} = (1)(2) + (9)( –2)

Entry for m_{22} = (1)( –3) + (9)(0)

Entry for m_{23} = (1)(1) + (9)(5)

The answer

matrix operations look like this:

M =

Matrix M simplifies to:

AB = M =

Notice that if you were to try to do BA, you could not because the dimensions would not match up. Always check your dimensions before beginning any multiplication!