SAT II Math I : Matrices

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

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Example Question #42 : Matrices And Vectors

If , what is ?

Possible Answers:

Correct answer:

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 .

Possible Answers:

 does not exist.

Correct answer:

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:  

Possible Answers:

Correct answer:

Explanation:

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:

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?

Possible Answers:

None of these

Correct answer:

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?

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 

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?

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 

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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