### All Precalculus Resources

## Example Questions

### Example Question #1 : Find The Product Of Two Matrices

Find .

**Possible Answers:**

No Solution

**Correct answer:**

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 3x* 3 and 3*x1).

**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 .

**Possible Answers:**

No Solution

**Correct answer:**

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 .

**Possible Answers:**

No Solution

**Correct answer:**

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

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 .

**Possible Answers:**

No Solution

**Correct answer:**

No Solution

The dimensions of A and B are as follows: A= 3x**1**, B= **2**x3

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.

The answer is **No Solution**.

### Example Question #5 : Find The Product Of Two Matrices

We consider the matrix equality:

Find the that makes the matrix equality possible.

**Possible Answers:**

There is no that satisfies the above equality.

**Correct answer:**

There is no that satisfies the above equality.

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 #6 : Find The Product Of Two Matrices

Let be the matrix defined by:

The value of ( the nth power of ) is:

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

The correct answer is itself.

**Correct answer:**

The correct answer is itself.

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?

**Possible Answers:**

**Correct answer:**

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 #9 : 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 ?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

Note that when we multiply the first row by the first colum we get: ( times), this gives the value of .

All other rows are zeros, and therefore we have zeros in the other entries.

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