### All SAT II Math II Resources

## Example Questions

### Example Question #3 : Matrices

Define matrix

For which of the following matrix values of is the expression defined?

**Possible Answers:**

The expression is defined for all of the values of given in the other responses.

**Correct answer:**

For the matrix product to be defined, itis necessary and sufficent for the number of columns in to be equal to the number of rows in .

has two columns. Of the choices, only

has two rows, making it the correct choice.

### Example Question #4 : Matrices

Calculate:

**Possible Answers:**

**Correct answer:**

To subtract two matrices, subtract the elements in corresponding positions:

### Example Question #5 : Matrices

Evaluate:

**Possible Answers:**

**Correct answer:**

The determinant of the matrix is

.

Substitute :

### Example Question #6 : Matrices

Give the determinant of the matrix

**Possible Answers:**

**Correct answer:**

The determinant of the matrix is

.

Substitute , :

### Example Question #7 : Matrices

Multiply:

**Possible Answers:**

**Correct answer:**

The product of a 2 x 2 matrix and a 2 x 1 matrix is a 2 x 1 matrix.

Multiply each row in the first matrix by the column matrix by multiplying elements in corresponding positions, then adding the products, as follows:

### Example Question #8 : Matrices

Let .

Give .

**Possible Answers:**

is not defined.

**Correct answer:**

is not defined.

has three rows and two columns; since the number of rows is not equal to the number of columns, is not a square matrix, and, therefore, it does not have an inverse.

### Example Question #9 : Matrices

Define matrix .

For which of the following matrix values of is the expression defined?

I:

II:

III:

**Possible Answers:**

I, II, and III

I and II only

II and II only

I only

I and III only

**Correct answer:**

I only

For the matrix sum to be defined, it is necessary and sufficent for and to have the same number of rows *and* the same number of columns. has three rows and two columns; of the three choices, only (I) has the same dimensions.

### Example Question #10 : Matrices

Let and be the 2 x 2 identity matrix.

Let .

Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

The 2 x 2 identity matrix is .

, or, equivalently,

,

so

Subtract the elements in the corresponding positions:

### Example Question #11 : Matrices

Calculate:

**Possible Answers:**

**Correct answer:**

To add two matrices, add the elements in corresponding positions:

### Example Question #12 : Matrices

Solve for :

**Possible Answers:**

or

or

The equation has no solution.

**Correct answer:**

or

The determinant of a matrix can be evaluated as follows:

Therefore, the equation can be rewritten:

The solution set is

or .

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