### All Linear Algebra Resources

## Example Questions

### Example Question #61 : Matrix Matrix Product

If , then which expression is equal to ?

**Possible Answers:**

**Correct answer:**

The transpose of a column matrix is the row matrix with the same entries, so

, so

Find by simply adding the products of corresponding entries:

So

Apply the trigonometric identity

,

setting :

### Example Question #62 : Matrix Matrix Product

is a matrix. is a matrix. and are both nonsingular.

Which expression is defined?

**Possible Answers:**

**Correct answer:**

and , being nonsquare matrices, cannot have inverses, so can be eliminated as a choice.

The product of a matrix and a matrix is a matrix. Therefore:

is a matrix; and are also matrices.

Similarly,

, , and are matrices.

Matrices of different dimensions cannot be added, so , , and can be eliminated.

and are and matrices, respectively. Therefore, is a matrix; , the sum of matrices, is a defined expression.

### Example Question #63 : Matrix Matrix Product

If , then which expression is equal to ?

**Possible Answers:**

**Correct answer:**

The transpose of a column matrix is the row matrix with the same entries, so if

,

then

Find by simply adding the products of corresponding entries:

Apply the trigonometric identity

Setting ,

.

### Example Question #64 : Matrix Matrix Product

Find .

**Possible Answers:**

None of the other choices gives the correct response.

**Correct answer:**

None of the other choices gives the correct response.

is a diagonal matrix. The product of two diagonal matrices can be found by multiplying elements in corresponding diagonal positions; the idea can be extended to powers of matrices, so

By DeMoivre's Theorem,

Set and in the upper left element:

Set and in the lower right element:

Therefore,

,

which is not among the choices.

### Example Question #65 : Matrix Matrix Product

Which of the following is equal to ?

**Possible Answers:**

None of the other choices gives the correct response.

**Correct answer:**

An easy way to find this is to note that ; therefore, we can find by squaring and squaring the result.

Matrix multiplication is worked row by column - each row in the former matrix is multiplied by each column in the latter by adding the products of elements in corresponding positions, as follows:

Now square this:

### Example Question #66 : Matrix Matrix Product

Evaluate .

**Possible Answers:**

None of the other choices gives the correct response.

**Correct answer:**

is a diagonal matrix, so it can be raised to a power by raising the individual entries to that power:

By DeMoivre's Theorem,

.

We will need to rewrite the entries in the matrix as sums, not differences. We can do this by noting that the cosine and sine functions are even and odd, respectively, so can be rewritten as

Applying DeMoivre's Theorem,

The coterminal angle for both and is , so

### Example Question #61 : Matrix Matrix Product

is a column matrix with seven entries. Which of the following is true of ?

**Possible Answers:**

is a row matrix with seven entries.

is a matrix with a single entry.

is a scalar.

is a column matrix with seven entries.

is a matrix.

**Correct answer:**

is a matrix with a single entry.

, a column matrix with ten entries, is a matrix; , its transpose, is a matrix.

If and are and matrices, respectively, then the product is a matrix. Therefore, is a matrix - a matrix with a single entry.

### Example Question #68 : Matrix Matrix Product

is a column matrix with ten entries. Which of the following is true of ?

**Possible Answers:**

is a scalar.

is a matrix.

is a matrix with a single entry.

is a column matrix with ten elements.

is a row matrix with ten elements.

**Correct answer:**

is a matrix.

, a column matrix with ten entries, is a matrix; , its transpose, is a matrix.

If and are and matrices, respectively, then the product is a matrix. Therefore, is a matrix.

### Example Question #69 : Matrix Matrix Product

Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

and are both elementary matrices, in that each can be formed from the (four-by-four) identity matrix by a single row operation.

Since differs from in that the entry is in Row 2, Column 2, the row operation is . Since differs from in that entry is in Row 3, Column 2, the row operation is .

Premultiplying a matrix by an elementary matrix has the effect of performing that row operation on the matrix. Looking at as :

Premultiply by by performing the operation :

Premultiply by by performing the operation :

This is the correct product.

### Example Question #70 : Matrix Matrix Product

refers to the identity matrix.

. Which of the following matrices could be equal to ?

**Possible Answers:**

None of the other responses gives a correct answer.

**Correct answer:**

All of the matrices are diagonal, so the seventh power of each can be determined by simply taking the seventh power of the individual entries in the main diagonal. Also, note that each entry in each choice is of the form

.

By DeMoivre's Theorem, for any real ,

Combining these ideas, we can take the seventh power of each matrix and determine which exponentiation yields the identity.

If

,

then

If

,

then

If

,

then

If

,

then

is the only possible matrix value of among the choices.

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