Matrix Multiplication

Before attempting matrix multiplication, makes sure you know the basics first.

Multiplication is much more complicated than some of the other matrix operations, like matrix addition and scalar multiplication. GradeA will show you two approaches: the Turn & Flip and the Zipper. Choose the method you like the best!

Before you can multiply matrices, you need to know when the operation is possible. Sometimes you cannot multiply the two matrices together at all.


When is Matrix Multiplication Defined?

The key to answering this question is to look at the dimensions of each matrix. Remember, they are always listed as row x column.

To see if the operation is defined, list the two dimensions next to each other. Important Note: The order does matter!

Next, check to see if the middle numbers are the same!

  Example #1 Example #2
  Matrix Dimensions

Once the matrix multiplication is defined, you can find the dimensions of the result (the answer). The outside numbers will give you the new dimensions.


For the examples above, the results would be (2 x 3) and (3 x 1).

Here are more examples:

Original Dimensions:
(3 x 1)(3 x 1)
(2 x 3)(3 x 2)
(4 x 1)(2 x 4)
Multiplication Possible? No! (different) Yes! (same) No! (different)
Resulting Dimensions:
(2 x 2)

The Turn and Flip Method

Now that you know how to determine the dimensions of the resulting matrix, now you need to know how to actually multiply them.

  Example: Matrix Multiplication

We are multiplying a (2 x 2)(2 x 2) - so the result will also be a (2 x 2).

You will need to multiply each corresponding row times the corresponding column - notice the circles below. Once you multiply the matching pieces, add the results.

Matrix Multiplication:  Turn & Flip Method

It might seem a little confusing at first, but after a little practice you will get it. Of course, you might prefer to use the zipper method instead...

The Zipper Method

If you had a difficult time understanding the turn and flip method, maybe the zipper method will be easier for you to understand.

We will use the same example:

Matrix Multiplication

First, take the second matrix and raise it above the first.


Zipper Method Setup

Now you are going to "zip" in the numbers that line up. Look at the light blue arrows, as well as the colored circles...

Matrix Multiplication: Zipper Method

So, now that you have seen both methods of matrix multiplication, which do you prefer: the turn and flip method or the zipper method?

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