### All SAT II Math I Resources

## Example Questions

### Example Question #42 : Matrices And Vectors

If , what is ?

**Possible Answers:**

**Correct answer:**

Begin by distributing the fraction through the matrix on the left side of the equation. This will simplify the contents, given that they are factors of :

Now, this means that your equation looks like:

This simply means:

and

or

Therefore,

### Example Question #11 : Matrices

Let and .

Evaluate .

**Possible Answers:**

does not exist.

**Correct answer:**

The inverse of any two-by-two matrix can be found according to this pattern:

If

then

,

where determinant is equal to .

Therefore, if , then , the second row/first column entry in the matrix , can be found by setting , then evaluating:

.

### Example Question #11 : Matrices

Solve:

**Possible Answers:**

**Correct answer:**

To compute the matrices, simply add the terms with the correct placement in the matrices. The resulting matrix is two by two.

The answer is:

### Example Question #11 : Matrices

.

**Possible Answers:**

**Correct answer:**

A matrix lacks an inverse if and only if its determinant is equal to zero. The determinant of is

Set this equal to 0 and solve for :

,

the correct response.

### Example Question #12 : Matrices

Let

Which of the following values of makes a matrix without an inverse?

**Possible Answers:**

None of these

**Correct answer:**

A matrix lacks an inverse if and only if its determinant is equal to zero. The determinant of is

.

We seek the value of that sets this quantity equal to 0. Setting it as such then solving for :

,

the correct response.

### Example Question #55 : Mathematical Relationships

Let equal the following:

Which of the following values of makes a matrix without an inverse?

**Possible Answers:**

There is one such value:

There is one such value:

There are two such values: or

There are two such values: or

There are two such values: or

**Correct answer:**

There are two such values: or

A matrix lacks an inverse if and only if its determinant is equal to zero. The determinant of is

Setting this equal to 0 and solving for :

### Example Question #54 : Mathematical Relationships

Let equal the following:

.

Which of the following real values of makes a matrix without an inverse?

**Possible Answers:**

There are two such values: or

has an inverse for all real values of

There are two such values: or

There are two such values: or

There is one such value:

**Correct answer:**

has an inverse for all real values of

A matrix lacks an inverse if and only if its determinant is equal to zero. The determinant of is

, so

Since the square of all real numbers is nonnegative, this equation has no real solution. It follows that the determinant cannot be 0 for any real value of , and that must have an inverse for all real .