Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : The Inverse

Study concepts, example questions & explanations for Linear Algebra

Create An Account

All Linear Algebra Resources

4 Diagnostic Tests 108 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #31 : The Inverse

is a singular matrix; is a nonsingular matrix.

Which is true of A B ?

Possible Answers:

A B must be a nonsingular matrix.

A B must be a singular matrix.

A B may be singular or nonsingular.

Correct answer:

A B must be a singular matrix.

Explanation:

A matrix is singular - that is, without an inverse - if and only if its determinant is 0. Since is a singular matrix and is a nonsingular matrix, $\det A = 0$ . The product of two matrices has as its determinant the product of the individual determinants, so

$\det AB = \det A \cdot \det B = 0 \cdot \det B = 0$ .

must be singular.

Report an Error

Example Question #31 : The Inverse

A matrix has as its determinant $\cos \frac{2\pi}{17}$ .

True or false: the determinant of $A^{-1}$ is $\csc \frac{2\pi}{17}$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

The determinant of the inverse $A^{-1}$ of matrix is the reciprocal of the determinant of , so

$\det A^{-1} = \frac{1}{\det A } = \frac{1}{\cos \frac{2 \pi}{17} } = \sec \frac{2 \pi}{17}$ .

The statement is false.

Report an Error

Example Question #255 : Operations And Properties

$A = \begin{bmatrix} 2^{a} & 1 \\1 & 2^{-a} \end{bmatrix}$

For which of the following values of is a singular matrix?

Possible Answers:

is singular regardless of the value of .

$a = \ln 2$

a = 1

a= 0

$a = \log 2$

Correct answer:

is singular regardless of the value of .

Explanation:

is singular - that is, it has no inverse - if and only if its determinant is equal to 0. the determinant of a $2 \times 2$ matrix

$\begin{bmatrix} a_{11} & a_{12 } \\ a_{21} & a_{22 } \end{bmatrix}$

is $\det A = a_{11} a_{22} - a_{12} a_{21}$

Substitute appropriately:

$\det A = 2^{a} \cdot 2^{-a}- 1 \cdot 1$

$= 2^{a -a}- 1$

$= 2^{0}- 1$

= 1 - 1

= 0

Regardless of the value of , has determinant 0 and is a singular matrix.

Report an Error

Example Question #32 : The Inverse

$A = \begin{bmatrix} a & 3 \\ 9 & -6 \end{bmatrix}$

Give so that is a singular matrix.

Possible Answers:

a=18

a=0

a=-18

a = -4.5

a = 4.5

Correct answer:

a = -4.5

Explanation:

$A = \begin{bmatrix} a & 3 \\ 9 & -6 \end{bmatrix}$

is a singular matrix - one without an inverse - if and only if its determinant is equal to 0. The determinant of a $2 \times 2$ matrix

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$

is equal to

$\det A = ad - bc$

Substituting appropriately and setting this quantity equal to 0, solve for :

a(-6)-3(9) = 0

-6a -27 = 0

-6a = 27

a = -4.5

Report an Error

Example Question #32 : The Inverse

A matrix has as its determinant $\cot \frac{13\pi}{15}$ .

True or false: $A^{-1 }$ has as its determinant $\tan \frac{13\pi}{15}$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

The determinant of the inverse $A^{-1}$ of matrix is the reciprocal of the determinant of , so

$\det A^{-1} = \frac{1}{\det A } = \frac{1}{\cot \frac{13 \pi}{15} } = \tan \frac{13 \pi}{15}$ .

The statement is true.

Report an Error

Example Question #34 : The Inverse

$A = \begin{bmatrix} a &b \\ c & d \end{bmatrix}$ such that $\det A = 4$ .

Which of the following is equal to $A^{-1}$ ?

Possible Answers:

$\begin{bmatrix} 4 a &-4 b \\ -4 c & 4d \end{bmatrix}$

$\begin{bmatrix} \frac{d }{4}&- \frac{c}{4} \\ \\ - \frac{b}{4} & \frac{a}{4} \end{bmatrix}$

$\begin{bmatrix} \frac{d }{4}&- \frac{b}{4} \\ \\ - \frac{c}{4} & \frac{a}{4} \end{bmatrix}$

$\begin{bmatrix} 4 d &-4 b \\ -4 c & 4a \end{bmatrix}$

$\begin{bmatrix} \frac{a }{4}&- \frac{b}{4} \\ \\ - \frac{d}{4} & \frac{c}{4} \end{bmatrix}$

Correct answer:

$\begin{bmatrix} \frac{d }{4}&- \frac{b}{4} \\ \\ - \frac{c}{4} & \frac{a}{4} \end{bmatrix}$

Explanation:

The inverse of a matrix $A = \begin{bmatrix} a &b \\ c & d \end{bmatrix}$ , if it exists, is

$A^{-1} =\frac{1}{\det A} \begin{bmatrix} d&- b \\- c &a \end{bmatrix}$

Since $\det A = 4$ ,

$A^{-1} =\frac{1}{4} \begin{bmatrix} d&- b \\- c &a \end{bmatrix}$

$=\begin{bmatrix} \frac{d }{4}&- \frac{b}{4} \\ \\ - \frac{c}{4} & \frac{a}{4} \end{bmatrix}$

Report an Error

Example Question #256 : Operations And Properties

$A = \begin{bmatrix} \cos \theta & \sin \theta \\ \cos (\frac{\pi}{2} - \theta )& \sin (\frac{\pi}{2}- \theta ) \end{bmatrix}$

For which of the following values of $\theta$ is a singular matrix?

Possible Answers:

$\theta = \frac{11 \pi}{3 }$

cannot be singular for any value of $\theta \;$ .

$\theta = \frac{11 \pi}{6}$

$\theta = \frac{11 \pi}{2 }$

$\theta = \frac{11 \pi}{4 }$

Correct answer:

$\theta = \frac{11 \pi}{4 }$

Explanation:

For any value of $\theta \;$ ,

$\cos\left ( \frac{\pi}{2} - \theta \right ) = \sin \theta$

and, equivalently,

$\sin \left ( \frac{\pi}{2} - \theta \right ) = \cos \theta$ .

can be rewritten as

$A = \begin{bmatrix} \cos \theta & \sin \theta \\ \sin \theta &\cos \theta \end{bmatrix}$

is singular - that is, it has no inverse - if and only if its determinant is equal to 0. The determinant of a $2 \times 2$ matrix

$\begin{bmatrix} a_{11} & a_{12 } \\ a_{21} & a_{22 } \end{bmatrix}$

is $\det A = a_{11} a_{22} - a_{12} a_{21}$

Substitute appropriately:

$\det A = \cos \theta \cdot \cos \theta - \sin \theta \cdot \sin \theta$

$= \cos ^{2} \theta - \sin ^{2} \theta$

$= \cos 2 \theta$

by a trigonometric identity.

Therefore, is singular if and only if

$\cos 2 \theta = 0$ .

It follows that for to be singular,

$2 \theta = \frac{\pi}{2} + n \pi = \frac{\pi+ 2n \pi }{2}$ for some integer value .

or, equivalently,

$\theta = \frac{\pi+ 2n \pi }{4}$

This can be restated:

$\theta = \frac{ (2n +1) \pi }{4}$ , or

$\theta = \frac{N \pi }{4}$ for an odd positive or negative integer .

The only choice that fits this description is $\theta = \frac{11 \pi}{4 }$ , which is the correct choice.

Report an Error

Example Question #33 : The Inverse

is an involutory matrix.

True or false: It follows that $A^{-1}$ is also an involutory matrix.

Possible Answers:

False

True

Correct answer:

True

Explanation:

A matrix is involutory if $A = A^{-1}$ . Since and $A^{-1}$ are the same matrix, it immediately follows that $A^{-1}$ is involutory.

Report an Error

Example Question #34 : The Inverse

A square matrix is invertible if and only if-

Possible Answers:

None of the other answers

it is diagonalizable

its determinant is .

it is similar to the zero matrix

Correct answer:

None of the other answers

Explanation:

All of these are criteria for a matrix NOT to be invertible, (except for diagonalization; some diagonalizable matrices are invertible and some aren't.)

Report an Error

Example Question #335 : Linear Algebra

$A = \begin{bmatrix} \frac{1}{2} & \frac{1}{3} \\ \\ \frac{2}{3} & \frac{1}{6}\end{bmatrix}$

Find $A^{-1}$ .

Possible Answers:

$\begin{bmatrix}\frac{18}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & \frac{6}{ 5} \end{bmatrix}$

$\begin{bmatrix}\frac{6}{ 5} &\frac{12}{ 5} \\ \\\frac{24}{ 5} & \frac{18}{ 5} \end{bmatrix}$

$\begin{bmatrix}-\frac{6}{ 5} &-\frac{12}{ 5} \\ \\- \frac{24}{ 5} & -\frac{18}{ 5} \end{bmatrix}$

$\begin{bmatrix}-\frac{18}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & -\frac{6}{ 5} \end{bmatrix}$

$\begin{bmatrix}-\frac{6}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & -\frac{18}{ 5} \end{bmatrix}$

Correct answer:

$\begin{bmatrix}-\frac{6}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & -\frac{18}{ 5} \end{bmatrix}$

Explanation:

The inverse of a two-by-two matrix

$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21}& a_{22} \end{bmatrix}$

is the matrix

$A^{-1} =\frac{1}{D} \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21}& a_{11} \end{bmatrix}$ .

is the determinant of the matrix, which is the product of the main diagonal elements minus the product of the other two:

$D = a_{11}a_{22}- a_{12}a_{21}$

$= \frac{1}{2} \left ( \frac{1}{6} \right ) - \frac{1}{3}\left ( \frac{2}{3} \right )$

$= \frac{1}{12} - \frac{2}{9}$

$= -\frac{5}{36}$

Therefore,

$A^{-1} =\frac{1}{ -\frac{5}{36}} \begin{bmatrix} \frac{1}{6} & -\frac{1}{3} \\ \\ -\frac{2}{3} & \frac{1}{2}\end{bmatrix}$

$=-\frac{36}{ 5} \begin{bmatrix} \frac{1}{6} & -\frac{1}{3} \\ \\ -\frac{2}{3} & \frac{1}{2}\end{bmatrix}$

$= \begin{bmatrix}-\frac{36}{ 5} \left ( \frac{1}{6} \right ) &-\frac{36}{ 5} \left ( -\frac{1}{3} \right ) \\ \\ -\frac{36}{ 5} \left ( -\frac{2}{3} \right ) & -\frac{36}{ 5} \left ( \frac{1}{2} \right ) \end{bmatrix}$

$= \begin{bmatrix}-\frac{6}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & -\frac{18}{ 5} \end{bmatrix}$

Report an Error

View Linear Algebra Tutors

Ying
Certified Tutor

Iowa State University, Master of Science, Mathematics.

View Linear Algebra Tutors

Scott
Certified Tutor

Florida State University, Bachelor of Science, Applied Mathematics. Florida State University, Bachelor of Science, Astrophysi...

View Linear Algebra Tutors

Yoonsik
Certified Tutor

Seoul National University, Bachelor of Science, Physics. University of Pennsylvania, Doctor of Philosophy, Atomic and Molecul...

All Linear Algebra Resources

4 Diagnostic Tests 108 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

LSAT Tutors in Los Angeles, Chemistry Tutors in Chicago, Calculus Tutors in Seattle, ISEE Tutors in Dallas Fort Worth, Statistics Tutors in Washington DC, MCAT Tutors in Phoenix, SAT Tutors in Houston, French Tutors in Chicago, ISEE Tutors in Phoenix, GMAT Tutors in Miami

Popular Courses & Classes

GRE Courses & Classes in Washington DC, MCAT Courses & Classes in Phoenix, ISEE Courses & Classes in New York City, ISEE Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Seattle, SAT Courses & Classes in Seattle, LSAT Courses & Classes in Washington DC, LSAT Courses & Classes in Boston, ACT Courses & Classes in Seattle, GMAT Courses & Classes in San Francisco-Bay Area

Popular Test Prep

SSAT Test Prep in Miami, SSAT Test Prep in Denver, ACT Test Prep in Dallas Fort Worth, ISEE Test Prep in Los Angeles, MCAT Test Prep in Atlanta, ISEE Test Prep in Atlanta, MCAT Test Prep in Washington DC, SSAT Test Prep in New York City, LSAT Test Prep in Seattle, GMAT Test Prep in Dallas Fort Worth

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Linear Algebra : The Inverse

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #31 : The Inverse

Example Question #31 : The Inverse

Example Question #255 : Operations And Properties

Example Question #32 : The Inverse

Example Question #32 : The Inverse

Example Question #34 : The Inverse

Example Question #256 : Operations And Properties

Example Question #33 : The Inverse

Example Question #34 : The Inverse

Example Question #335 : Linear Algebra

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: