### All Linear Algebra Resources

## Example Questions

### Example Question #21 : Matrices

and are both matrices with two rows and five columns.

Which of the following are defined?

(a)

(b)

(c)

(d)

**Possible Answers:**

(c) and (d) only

(a) and (b) only

All four of (a), (b), (c), and (d)

(a), (b) and (d) only

(a), (b) and (c) only

**Correct answer:**

(a) and (b) only

Two matrices can be added if and only if they have the same number of rows and the same number of columns. This is true of and , so and are defined.

For the product of two matrices to be defined, the number of columns in the first matrix must be equal to the number of rows in the second. cannot be defined, since has five columns and has two rows. cannot be defined for the same reason.

The correct choice is (a) and (b) only.

### Example Question #22 : Matrix Matrix Product

is a matrix with three rows and six columns. is a matrix with six rows and three columns.

Which of the following are defined?

(a)

(b)

(c)

(d)

**Possible Answers:**

(a) and (b) only

All four of (a), (b), (c), and (d)

(c) and (d) only

(a), (b) and (c) only

(a), (b) and (d) only

**Correct answer:**

(c) and (d) only

Two matrices can be added if and only if they have the same number of rows and the same number of columns. This is not true of and , so and are undefined.

For the product of two matrices to be defined, the number of columns in the first matrix must be equal to the number of rows in the second. has six columns and has six rows, so can be defined. has three columns and has three rows, so can be defined.

### Example Question #702 : Linear Algebra

Give the values of , , and so that the matrix

is a stochastic matrix (Markov chain).

**Possible Answers:**

The matrix cannot be made into a stochastic matrix regardless of the values of the variables.

**Correct answer:**

A stochastic matrix is a matrix of probabilities in which the entry is the probability that, given the fact that a given system is in a state , the system will be in state next. As such, the elements in each column of , being the probabilities that the system will change from a given state to each other state, respectively, must add up to 1.

Therefore,

The correct choice is that .

### Example Question #23 : Matrix Matrix Product

and .

True or false: .

**Possible Answers:**

False

True

**Correct answer:**

False

It is not necessary to find and in order to prove that the statement is false.

For the product of two matrices to be defined, the number of columns in the first matrix must be equal to the number of rows in the second. The result of multiplying two matrices when the first matrix has rows and columns and the second has rows and columns is a matrix with rows and columns.

Therefore, since has two rows and three columns, and has three rows and two columns, it follows that has two rows and two columns, and has three rows and three columns. Since and have different dimensions, they cannot be equal.

### Example Question #25 : Matrix Matrix Product

Give the values of , , and so that the matrix

is an example of a stochastic matrix (Markov chain).

**Possible Answers:**

The matrix cannot be made into a stochastic matrix regardless of the values of the variables.

**Correct answer:**

The matrix cannot be made into a stochastic matrix regardless of the values of the variables.

A stochastic matrix is a matrix of probabilities in which the entry is the probability that, given the fact that a given system is in a state , the system will be in state next. As such, the elements in each column of , being the probabilities that the system will change from a given state to each other state, respectively, must add up to 1.

Therefore,

However, since a stochastic matrix is, by definition, a matrix of probabilities, each element of the matrix must fall in the interval . This rule has been violated, so no values of the variables can make the matrix a stochastic matrix.

### Example Question #21 : Matrix Matrix Product

Mary and Marty are playing a game whose board comprises two spaces, marked "X' and "Y", as shown above. For each turn, one player rolls a fair six-sided die, then moves to the space indicated based on where the player is already and what the player rolls. For example, if the player is on space "X" and rolls a "2", he moves to space "Y", since one of the numbers on the arrow going from "X" to 'Y" is "2"; similarly, if he is on space "X" and rolls a "6", he will stay where he is.

Give the stochastic matrix which represents the probabilities that, given that a player is on a particular space, he or she will be on each given space *two turns later.*

**Possible Answers:**

**Correct answer:**

A stochastic matrix is a matrix of probabilities in which the entry is the probability that, given the fact that a given system is in a state , the system will be in state next.

We will let State 1 be that the player is on Space "X" and State 2 be that the player is on Space "Y". If the player is on Space "X", the probability that he will still be on that space after one turn is , since one roll out of six will allow him to stay there; similarly, the probability that he will be on Space "Y" is . If a player is on Space "B", the same probabilities are, respectively, and . The stochastic matrix representing these probabilities is

.

The stochastic matrix representing the probabilities of being on a particular space after *two* terms is the square of this matrix , which is

The entry in column , row of the product is the product of row of and column of - the sum of the products of the numbers that appear in the corresponding positions of the row and the column. Therefore,

,

the correct choice.

### Example Question #21 : Matrix Matrix Product

and are both defined products. has four rows and three columns.

Which of the following is true about the number of rows and columns in ?

**Possible Answers:**

must have four rows and three columns.

must either have three rows and four columns or four rows and three columns.

must have three rows but can have any number of columns.

must have three rows and four columns.

can have any number of rows but must have four columns.

**Correct answer:**

must have three rows and four columns.

For a matrix product to be defined, it must hold that the number of rows in is equal to the number of columns in . has three columns, so must have three rows. For to be defined, by similar reasoning, must have four columns.

### Example Question #28 : Matrix Matrix Product

where

and

Evaluate .

(You may assume that both and have inverses. )

**Possible Answers:**

None of the other choices gives a correct response.

**Correct answer:**

While implies that , it is not necessary to do the calculations in order to find .

differs from only in its second row, in which each element is times the corresponding element in :

Therefore, is the result of a row operation on , namely,

Therefore, is the product of an elementary matrix and ; the elementary matrix for the given row operation is the one in which the operation is performed on the (four-by-four) identity matrix, which is

.

### Example Question #29 : Matrix Matrix Product

where

and

Evaluate .

(You may assume that both and have inverses. )

**Possible Answers:**

**Correct answer:**

While implies that , it is not necessary to do the calculations in order to find .

differs from only in that its first and third rows have reversed positions. Therefore, is the result of a row operation on , namely,

Therefore, is the product of an elementary matrix and ; the elementary matrix for the given row operation is the one in which the operation is performed on the (four-by-four) identity matrix, which is

.

### Example Question #21 : Matrix Matrix Product

Let , , and

True or false: is an example of a valid -factorization of .

**Possible Answers:**

False, because is not the right kind of matrix.

False, because is not a factorization of .

True

False, because is not the right kind of matrix.

**Correct answer:**

False, because is not the right kind of matrix.

An -factorization is a way of expressing a matrix as a product of two matrices and . For the factorization to be valid:

1) must be a **L**ower triangular matrix - all elements *above* its main diagonal (upper left corner to lower right corner) must be "0".

2) must be an **U**pper triangular matrix - all elements *below *its main diagonal must be "0".

3)

can be seen to have a nonzero element above its main diagonal - the 4 in Row 1, Column 2. Consequently, is not a lower triangular matrix. This makes invalid as an -factorization.