Linear Algebra : Matrix-Matrix Product

Example Questions

Example Question #41 : Matrix Matrix Product

Find the product .

Possible Answers:

cannot be multiplied

Correct answer:

Explanation:

If  is an  matrix and  is an  matrix,

can only be multiplied if , or the number of columns in  equals the number of rows in .  Otherwise there is a mismatch, and the two matrices can not be multiplied.   will be an  matrix

Since  is a  matrix and  is a  matrix, then  can be multiplied and will have the dimensions

To find the product , you must find the dot product of the rows of  and the columns of

We find  by finding the dot product of the row  of  and column  of .

We find  by finding the dot product of the row  of  and column  of .

We use the same method to find  the rest of the matrix values

Example Question #42 : Matrix Matrix Product

Find the product .

Possible Answers:

cannot be multiplied.

Correct answer:

cannot be multiplied.

Explanation:

If  is an  matrix and  is an  matrix,

can only be multiplied if , or the number of columns in  equals the number of rows in .  Otherwise there is a mismatch, and the two matrices can not be multiplied.   will be an  matrix

Since  is a  matrix and  is a  matrix,  so  cannot be multiplied.

Example Question #43 : Matrix Matrix Product

Find the product .

Possible Answers:

cannot be multiplied

Correct answer:

Explanation:

If  is an  matrix and  is an  matrix,

can only be multiplied if , or the number of columns in  equals the number of rows in .  Otherwise there is a mismatch, and the two matrices can not be multiplied.   will be an  matrix

Since  is a  matrix and  is a  matrix, then  can be multiplied and will have the dimensions

To find the product , you must find the dot product of the rows of  and the columns of

We find  by finding the dot product of the row  of  and column  of .

We find  by finding the dot product of the row  of  and column  of .

We use the same method to find  the rest of the matrix values

Example Question #44 : Matrix Matrix Product

Find the product .

Possible Answers:

cannot be multiplied

Correct answer:

Explanation:

If  is an  matrix and  is an  matrix,

can only be multiplied if , or the number of columns in  equals the number of rows in .  Otherwise there is a mismatch, and the two matrices can not be multiplied.   will be an  matrix

Since  is a  matrix and  is a  matrix, then  can be multiplied and will have the dimensions

To find the product , you must find the dot product of the rows of  and the columns of

We find  by finding the dot product of the row  of  and column  of .

We find  by finding the dot product of the row  of  and column  of .

We use the same method to find  the rest of the matrix values

Example Question #45 : Matrix Matrix Product

Find the product .

Possible Answers:

cannot be multiplied.

Correct answer:

cannot be multiplied.

Explanation:

If  is an  matrix and  is an  matrix,

can only be multiplied if , or the number of columns in  equals the number of rows in .  Otherwise there is a mismatch, and the two matrices can not be multiplied.   will be an  matrix

Since  is a  matrix and  is a  matrix,  so  cannot be multiplied.

Example Question #41 : Matrices

True or false:

is an example of an idempotent matrix.

Possible Answers:

True

False

Correct answer:

True

Explanation:

is an idempotent matrix, by definition, if . Multiply  by itself by multiplying rows by columns - multiplying elements in corresponding positions and adding the products:

.

, making  idempotent.

Example Question #47 : Matrix Matrix Product

True or false:

is an example of an idempotent matrix.

Possible Answers:

False

True

Correct answer:

False

Explanation:

is an idempotent matrix, by definition, if . Multiply  by itself by multiplying rows by columns - multiplying elements in corresponding positions and adding the products:

, so  is not idempotent.

Example Question #42 : Matrices

For any given value , how many  nonsingular idempotent matrices exist?

Possible Answers:

Two

Infinitely many

Zero

One

Correct answer:

One

Explanation:

is nonsingular, by definition, if it has an inverse - that is, if  exists.  is an idempotent matrix, by definition, if

Premultiplying both sides of the equation by , we get

,

where  is the  identity matrix.

Matrix multiplication is associative, so

.

Therefore, the only nonsingular idempotent matrix of a given dimension  is the identity.

Example Question #43 : Matrices

The above diagram shows a board for a game of chance. A player moves according to the flip of a fair coin, depending on his current location. For example, if he is on the green square, he will move to the orange square if the coin comes up heads, and to the pink square if it comes up tails.

Construct a stochastic matrix that models this game, with the rows/columns representing, in order, the orange, pink, blue, and green squares.

Possible Answers:

Correct answer:

Explanation:

Since the coin is fair, each outcome will come up with probability 0.5. Therefore, a player on orange end up on orange or pink with 0.5 probability each; a player on pink will end up on orange or blue with probability 0.5 each;  a player on blue will end up on pink or green with probability 0.5 each; and a player on green will end up on orange or pink with probability 0.5 each. The matrix that models this is

.

Example Question #41 : Matrices

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

The above diagram shows a board for a game of chance. A player moves according to the flip of a fair coin, depending on his current location. For example, if he is on the green square, he will move to the orange square if the coin comes up heads, and to the pink square if it comes up tails.

The player agrees to start on the pink square. Which square is he most likely to end up on after eight moves?

Possible Answers:

Orange

Pink

Green

Blue

Correct answer:

Orange

Explanation:

Since the coin is fair, each outcome will come up with probability 0.5. Therefore, a player on orange end up on orange or pink with 0.5 probability each; a player on pink will end up on orange or blue with probability 0.5 each;  a player on blue will end up on pink or green with probability 0.5 each; and a player on green will end up on orange or pink with probability 0.5 each. The matrix that models this is

.

The matrix that models the probabilities that the player will end up on a given square, given that he starts on a particular square, is , which through calculation is

Since the player is starting on pink, examine the second column; the greatest entry is in the second row, which represents ending up on orange.