Precalculus : Find the Product of Two Matrices

Example Questions

Example Question #1 : Find The Product Of Two Matrices

Find

No Solution

Explanation:

The dimensions of A and B are as follows: A= 3x3, B= 3x1

When we mulitply two matrices, we need to keep in mind their dimensions (in this case 3x3 and 3x1).

The two inner numbers need to be the same. Otherwise, we cannot multiply them. The product's dimensions will be the two outer numbers: 3x1.

Example Question #2 : Find The Product Of Two Matrices

Find

No Solution

Explanation:

The dimensions of both A and B are 2x2. Therefore, the matrix that results from their product will have the same dimensions.

Thus plugging in our values for this particular problem we get the following:

Example Question #3 : Find The Product Of Two Matrices

Find .

No Solution

Explanation:

The dimensions of A and B are as follows: A=1x3, B= 3x1.

Because the two inner numbers are the same, we can find the product.

The two outer numbers will tell us the dimensions of the product: 1x1.

Therefore, plugging in our values for this problem we get the following:

Example Question #4 : Find The Product Of Two Matrices

Find .

No Solution

No Solution

Explanation:

The dimensions of A and B are as follows: A= 3x1, B= 2x3

In order to be able to multiply matrices, the inner numbers need to be the same. In this case, they are 1 and 2. As such, we cannot find their product.

Example Question #5 : Find The Product Of Two Matrices

We consider the matrix equality:

Find the  that makes the matrix equality possible.

There is no  that satisfies the above equality.

There is no  that satisfies the above equality.

Explanation:

To have the above equality we need to have and .

means that , or . Trying all different values of , we see that no  can satisfy both matrices.

Therefore there is no  that satisfies the above equality.

Example Question #5 : Find The Product Of Two Matrices

Let  be the matrix defined by:

The value of ( the nth power of ) is:

Explanation:

We will use an induction proof to show this result.

We first note the above result holds for n=1. This means

We suppose that   and we need to show that:

By definition . By inductive hypothesis, we have:

Therefore,

This shows that the result is true for n+1. By the principle of mathematical induction we have the result.

Example Question #7 : Find The Product Of Two Matrices

We will consider the 5x5 matrix  defined by:

what is the value of ?

Explanation:

Note that:

Since .

This means that

Example Question #8 : Find The Product Of Two Matrices

Let  have the dimensions of a  matrix and  a  matrix. When is  possible?

Explanation:

We know that to be able to have the product of the 2 matrices, the size of the column of A must equal the size of the row of B. This gives :

.

Solving for n, we find

Since n is a natural number  is  the only possible solution.

Example Question #6 : Find The Product Of Two Matrices

We consider the matrices  and  below. We suppose that  and  are of the same size

What is the product ?

Explanation:

Note that every entry of the product matrix is the sum of  ( times) .

This gives  as every entry of the product of the two matrices.

Example Question #10 : Find The Product Of Two Matrices

We will consider the two matrices

We suppose that  and  have the same size

What is ?