Matrix Multiplication: Sum & Product

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Find where
, , and
Also demonstrate the distributive property of matrices.

Should we multiply or add first?

Since matrices have the distributive property, we'll eventually do both.

State the distributive property.

We'll be talking about rows and columns, so let's be sure which is which. Is a row left-right or up-and-down?

Left-right.

Let's do the addition first. Will be a number or a matrix?

A matrix.

In order to add two matrices, what is the requirement for their rows and columns?

Each matrix must have the same number of rows and each must have the same number of columns. It is not necessary for these two numbers to be the same, which is the case here.

Set up .

Simplify this.

Set up the multiplication of this result by y.

Simplify.

We get

Let's check by distributing first and then evaluating .
First, set up

Simplify.

We get

Next, set up

Simplify.

We get

Combine these results.

What can we conclude here?

We have demonstrated the distribution property of matrices:

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General Contents

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