Linear Algebra : Matrix-Matrix Product

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #71 : Matrix Matrix Product

Which of the following is equal to  ?

Possible Answers:

Correct answer:

Explanation:

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

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

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

 Postmultiply  by  by performing the operation :

Postmultiply  by  by performing the operation :

.

This is the correct product.

Example Question #72 : Matrix Matrix Product

Compute , where 

Possible Answers:

Not Possible

Correct answer:

Explanation:

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together. To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix. For this example, our product matrix will have dimensions of (2x2). Next, we notice that matrix A is the identity matrix. Any matrix multiplied by the identity matrix remains unchanged.  

Example Question #73 : Matrix Matrix Product

 is a stochastic matrix for a system. Evaluate .

Possible Answers:

Correct answer:

Explanation:

Each column of a stochastic matrix has entries whose sum is 1. For each column in , set the sum of the entries equal to 1; it is easiest to work with Columns 1, 2, and 3 in that order, for reasons that become apparent:

Column 1: 

Column 2: 

It is known that , so substitute:

Column 3: 

It is known that , so substitute:

Example Question #74 : Matrix Matrix Product

Calculate .

Possible Answers:

Correct answer:

Explanation:

This is not as daunting a task as it seems. 

First, find :

Multiply rows of  by columns of  by adding the products of corresponding elements, as follows:

Find  similarly:

This suggests a pattern:

 for any .

This can be confirmed by mathematical induction as follows:

Suppose .

Then 

Setting :

.

Example Question #75 : Matrix Matrix Product

In terms of  and , give a necessary and sufficient condition for  to be an involutory matrix.

Possible Answers:

 cannot be an involutory matrix regardless of the values of the variables.

Correct answer:

Explanation:

A matrix  is involutory if , or, equivalently, if . Multiply  by itself by multiplying rows by columns - add the products of corresponding elements - as follows:

If we set this equal to , the equation is

The only condition required to set these matrices equal is

,

or

,

making this condition necessary and sufficient for  to be involutory.

Example Question #76 : Matrix Matrix Product

Game

It is recommended that you use a calculator with matrix arithmetic capability for this exercise. 

Above is the board for a game.

A player may start at any space. He then rolls one of four polyhedral dice, depending on the space he is on:

If he is on the pink circle, he will roll a fair six-sided die.

If he is on the blue circle, he will roll a fair eight-sided die.

If he is on the orange circle, he will roll a fair ten-sided die.

If he is on the blue circle, he will roll a fair twelve-sided die.

Each side of each die is numbered with a different whole number, beginning with 1 and increasing incrementally (i.e. the twelve-sided die will be marked with the whole numbers 1-12). The player will move according to the following rules:

1) If he rolls a 1, he will move to (or stay on) the pink circle.

2) If he rolls a prime number, he will move one space clockwise.

3) If he rolls a composite number, he will move one space counterclockwise.

After a large enough number of turns, on which circle will the player tend to spend the most time?

Possible Answers:

Pink

Orange

Yellow

Blue

Correct answer:

Pink

Explanation:

A stochastic matrix for the game can be formed from the probability that the player will land on each space given the space he is on currently.

If he is on pink, he will roll a six-sided die; his probabilities are:

Pink:  (1 only)

Blue:  (2, 3, or 5)

Yellow:  (4 or 6)

 

If he is on blue, he will roll an eight-sided die; his probabilities are:

Pink:  (1, 4, 6, 8)

Orange:  (2, 3, 5, 7)

 

If he is on orange, he will roll a ten-sided die; his probabilities are:

Pink:  (1 only)

Blue:  (4, 6, 8, 9, 10)

Yellow:  (2, 3, 5, 7)

 

If he is on yellow, he will roll a twelve-sided die; his probabilities are:

Pink:  (1, 2, 3, 5, 7, 11)

Orange:  (4, 6, 8, 9, 10, 12 )

With the rows/columns representing pink, blue, orange, and yellow states, in that order, the stochastic matrix is 

Any stochastic matrix has 1 as its dominant eigenvalue; the eigenvector corresponding to that eigenvalue is its steady-state vector. We can find this vector directly, but we can also calculate it numerically.

The matrix that shows the probabilities after  turns is . If we raise  to a sufficiently large power - say,  - we see that the result approaches the matrix , where 

This is the steady state vector of the system. The greatest value appears in the first row, which represents the pink circle. It follows that a player will tend to spend most of his time on the pink circle as the game progresses.

Example Question #77 : Matrix Matrix Product

Evaluate .

Possible Answers:

Correct answer:

Explanation:

This is not as daunting a task as it seems. 

First, find .

Multiply rows of  by columns of  by adding the products of corresponding elements, as follows:

Note that 

It easily follows that

 

 can be calculated as follows:

Example Question #78 : Matrix Matrix Product

Game

It is recommended that you use a calculator with matrix arithmetic capability for this exercise. 

Above is the board for a game.

A player may start at any space. He then rolls one of four polyhedral dice, depending on the space he is on:

If he is on the pink circle, he will roll a fair six-sided die.

If he is on the blue circle, he will roll a fair eight-sided die.

If he is on the orange circle, he will roll a fair ten-sided die.

If he is on the blue circle, he will roll a fair twelve-sided die.

Each side of each die is numbered with a different whole number, beginning with 1 and increasing incrementally (i.e. the twelve-sided die will be marked with the whole numbers 1-12). The player will move according to the following rules:

1) If he rolls a 1, he will move to the opposite circle (where blue and yellow are opposite circles, and pink and orange are opposite circles).

2) If he rolls a prime number, he will move one space clockwise.

3) If he rolls a composite number, he will move one space counterclockwise.

After a large enough number of turns, on which circle will the player tend to spend the most time?

Possible Answers:

Orange

Blue

Yellow

Pink 

Correct answer:

Orange

Explanation:

A stochastic matrix for the game can be formed from the probability that the player will land on each space given the space he is on currently.

If he is on pink, he will roll a six-sided die; his probabilities are:

Blue:  (2, 3, or 5)

Orange:  (1 only)

Yellow:  (4 or 6)

 

If he is on blue, he will roll an eight-sided die; his probabilities are:

Pink:  (4, 6, 8)

Orange:  (2, 3, 5, 7)

Yellow:  (1 only)

 

If he is on orange, he will roll a ten-sided die; his probabilities are:

Pink:  (1 only)

Blue:  (4, 6, 8, 9, 10)

Yellow:  (2, 3, 5, 7)

 

If he is on yellow, he will roll a twelve-sided die; his probabilities are:

Pink:  (2, 3, 5, 7, 11)

Blue:  (1 only)

Orange:  (4, 6, 8, 9, 10, 12)

With the rows/columns representing pink, blue, orange, and yellow states, in that order, the stochastic matrix is 

Any stochastic matrix has 1 as its dominant eigenvalue; the eigenvector corresponding to that eigenvalue is its steady-state vector. We can find this vector directly, but we can also calculate it numerically.

The matrix that shows the probabilities after  turns is . If we raise  to a sufficiently large power - say,  - we see that the result approaches the matrix , where 

This is the steady state vector of the system. We see that the greatest value appears in the third row, which represents the orange circle. It follows that a player will tend to spend most of his time on the orange circle as the game progresses.

Example Question #79 : Matrix Matrix Product

Is  involutory, idempotent, or neither? 

Possible Answers:

Involutory

Idempotent

Neither

Correct answer:

Involutory

Explanation:

 is involutory if ; it is idempotent if . Therefore, to determine which, if either, describes , square . This can be done by multiplying rows of  by columns of  - adding the products of corresponding entries, as follows:

 is therefore involutory. 

Example Question #80 : Matrix Matrix Product

True or false:  is an example of an involutory matrix.

Possible Answers:

False

True

Correct answer:

True

Explanation:

 is an involutory matrix if , so square  to answer this question. 

This matrix can be squared more easily by noting that this matrix can be blocked as:

,

Where , and  is the  zero matrix. 

Now, find the squares of the two blocks:

, making  involutory.

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