Matrix Multiplication

We will illustrate matrix multiplication or matrix product by the following example.

Example:

Find C = A × B

Solution:

Step 1 : Multiply the elements in the first row of A with the corresponding elements in the first column of B. Add the products to get the element C ₁₁

Showing Step 1 in detail:

Step 2 : Multiply the elements in the first row of A with the corresponding elements in the second column of B. Add the products to get the element C ₁₂

Step 3 : Multiply the elements in the second row of A with the corresponding elements in the first column of B. Add the products to get the element C ₂₁

Step 4 : Multiply the elements in the second row of A with the corresponding elements in the second column of B. Add the products to get the element C ₂₂

Not all matrices can be multiplied together.

For example, the product of A and B is not defined.

We cannot multiply A and B because there are 3 elements in the row to be multiplied with 2 elements in the column

This means that we can only multiply two matrices if the number of elements in the row of the first matrix is equal to the number of elements in the column of the second matrix.

An easy method to determine whether two matrices can be multiplied together would be to check the order of the matrices.

Can we multiply matrix A and matrix C?

We can also know the order of the product.

Working it out we can see that

Checking the orders of the matrices will also help you to make sure that you multiplied the elements in the correct way.

Take note that matrix multiplication is not commutative that is

A × B ≠ B × A

The following video shows another example of matrix multiplication.

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