Multiplying a Matrix by Another Matrix

Unlike adding and subtracting matrices, when we multiply matrices, we don't just multiply the corresponding values. When multiplying matrices, we have to multiply each row by each column.

If $A$ is an $n$ by $r$ matrix, and $B$ is an $r$ by $m$ matrix then $A B$ is defined as $$(AB)_{ij}=\sum_{k=1}^r A_{ik} B_{kj}.$$

Let's look at an example:

Suppose we want to multiply the two matrices: We multiply each column by each row in the following way. The number in the $i$th row and $j$th column in the resulting matrix is obtained by multiplying each number in the $i$th row of the first matrix by the $j$th column of the second matrix, and then taking the sum.

So, we take the first row and first column to get the first value: Then, we proceed similarly:

So, we have

Note that the number of rows in the first matrix must be equal to the number of columns in the second matrix.

(Video) Multiplying Matrices

by ThinkwellVids

Here's a good video by ThinkwellVids showing you how to multiply 2 matrices together.

Now that we've learned the basics of the row and column matrix, we can move forward into more complicated types of matrix multiplication.

Multiplying Two Matrices

In this next lesson in the series of matrix multiplication, we will learn how to multiple two matrices together. This is also known as finding the dot product of matrices.

With matrix multiplication, order matters. That means that matrix multiplication is not commutative. Multiply the elements in the order in which the expression asks.

Before multiplying matrices, you must determine whether the product matrix is defined and what its dimensions will be.

There is one exception to note. Multiplying any matrix by the identity matrix leaves it unchanged.