# Linear Algebra : Matrix-Matrix Product

## Example Questions

### Example Question #91 : Matrix Matrix Product

is a  matrix.

is a  matrix.

is a  matrix.

is a  matrix.

is an  matrix.

Which of the following is a square matrix?

All four of the expressions given in the other choices are defined.

All four of the expressions given in the other choices are defined.

Explanation:

Two matrices can be multiplied if and only if the number of columns in the first is equal to the number of columns in the second. The number of rows in the first matrix and the number of columns in the second are the number of rows and columns in the product, respectively.

For example, the following matrices can be put together in order:

is a  matrix;  is a  matrix;  is defined and is a  matrix.

is a  matrix;  is an  matrix;  is defined and is a  matrix.

By extension, matrices can be linked in a product of three or more such that, if two matrices appear together, this same relation must hold. For example, since  is a  matrix,  is a  matrix, and  is an  matrix,  is defined and is a  matrix.

It follows by further extension that  is defined and is a  matrix, and that  is defined and is an  matrix.

is defined to be a  matrix. The transpose of the product of matrices is equal to the product of transposes in reverse, so

, a   matrix.

Similarly, , a  matrix.

The correct response is that all four given matrices are square.

### Example Question #92 : Matrix Matrix Product

is a singular  matrix.

is a nonsingular  matrix.

is a  matrix.

All of the following are undefined except:

None of the expressions given in the other choices are defined.

Explanation:

can be eliminated as a choice;  is not a square matrix, so the inverse of , does not exist.

can also be eliminated;  is a singular matrix, so, by definition,  does not exist.

Two matrices can be multiplied if and only if the number of columns in the first is equal to the number of columns in the second.  can be eliminated, since  has four columns and  has five rows.

The remaining choice is  is nonsingular, so  is defined. , like , is a  matrix;  is a  matrix, so its transpose  is ; thus,  is .  is , so  is .  is defined and is the correct choice.

### Example Question #93 : Matrix Matrix Product

Let  and .

Find .

is undefined.

Explanation:

First, it must be established that is defined. This is the case if and only if has as many columns as has rows. Since has two columns and has two rows, is defined.

Matrices are multiplied by multiplying each row of the first matrix by each column of the second - that is, by adding the products of the entries in corresponding positions. Thus,

### Example Question #94 : Matrix Matrix Product

Let and .

Find .

is undefined.

Explanation:

First, it must be established that is defined. This is the case if and only if has as many columns as has rows. Since has two columns and has two rows, is defined.

Matrices are multiplied by multiplying each row of the first matrix by each column of the second - that is, by adding the products of the entries in corresponding positions. Thus,

.

### Example Question #91 : Matrix Matrix Product

Let and .

Find .

is undefined.

is undefined.

Explanation:

First, it must be established that is defined. This is the case if and only if has as many columns as has rows. Since has two columns and has one row, is not defined.

### Example Question #96 : Matrix Matrix Product

and , where   and  stand for real quantities.

Which of the following must be a true statement?

None of the statements given in the other choices are correct.

Explanation:

Matrices are multiplied by multiplying each row of the first matrix by each column of the second - that is, by adding the products of the entries in corresponding positions. Thus,

If rows and columns are transposed in , it can be seen that

.

### Example Question #92 : Matrix Matrix Product

and , where   and  stand for real quantities.

Which of the following must be a true statement?

None of the statements given in the other choices are correct.

Explanation:

Matrices are multiplied by multiplying each row of the first matrix by each column of the second - that is, by adding the products of the entries in corresponding positions. Thus,

### Example Question #93 : Matrix Matrix Product

is a symmetric matrix.

True or false: It follows that is also a symmetric matrix.

True

False

False

Explanation:

This statement can be proved by counterexample.

Let .

is not symmetric, since its transpose,

is not equal to .

Then

Matrices are multiplied by multiplying each row of the first matrix by each column of the second - that is, by adding the products of the entries in corresponding positions. Thus,

.

Therefore, , the two-by-two identity, which is symmetric.

Since a nonsymmetric matrix exists whose square is symmetric, then the given statement is false.

### Example Question #99 : Matrix Matrix Product

Calculate .

is undefined.

is undefined.

Explanation:

Only square matrices can be taken to any power. Since is not a square matrix, having two rows and three columns, is undefined.

### Example Question #100 : Matrix Matrix Product

Calculate .

None of the other choices gives the correct response.

None of the other choices gives the correct response.

Explanation:

, the transpose of , can be found by transposing rows with columns.

, so

Matrices are multiplied by multiplying each row of the first matrix by each column of the second - that is, by adding the products of the entries in corresponding positions. Thus,

This is a square matrix, so it can be raised to a power. To raise a diagonal matrix to a power, simply raise each number in the main diagonal to that power:

.

This is not among the choices.