### All Linear Algebra Resources

## Example Questions

### Example Question #41 : Matrix Matrix Product

Find the product .

,

**Possible Answers:**

cannot be multiplied

**Correct answer:**

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.

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:**

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:**

If is an matrix and is 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.

If is an matrix and is 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

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

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

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:**

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

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.

Certified Tutor