# SAT II Math I : Matrices

## Example Questions

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### Example Question #42 : Matrices And Vectors

If , what is ?

Explanation:

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 .

does not exist.

Explanation:

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:

Explanation:

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

### Example Question #11 : Matrices

.

Explanation:

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?

None of these

Explanation:

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?

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

There are two such values:  or

Explanation:

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?

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:

has an inverse for all real values of

Explanation:

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 .

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