Linear Algebra : Matrix-Matrix Product

Study concepts, example questions & explanations for Linear Algebra

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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

Explanation:

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

Explanation:

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:

Explanation:

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

Explanation:

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.

Explanation:

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

Markov 1

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:

Explanation:

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.

Explanation:

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:

Explanation:

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:

Explanation:

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.

Explanation:

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 Lower triangular matrix - all elements above its main diagonal (upper left corner to lower right corner) must be "0".

2)  must be an Upper 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.

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