Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : Linear Algebra

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 #341 : Linear Algebra

$A= \begin{bmatrix} 0. 3 & 1. 7 \\ 0.5 & 2.3 \end{bmatrix}$

Find $A^{-1}$ .

Possible Answers:

$\begin{bmatrix} -14.375 &-10.625 \\- 3.125 &-1.875 \end{bmatrix}$

$\begin{bmatrix} -1.875 &-10.625 \\- 3.125 & -14.375 \end{bmatrix}$

$\begin{bmatrix} -14.375 &10.625 \\ 3.125 &-1.875 \end{bmatrix}$

$\begin{bmatrix} 14.375 &-10.625 \\- 3.125 & 1.875 \end{bmatrix}$

$\begin{bmatrix} -1.875 &10.625 \\ 3.125 & -14.375 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} -14.375 &10.625 \\ 3.125 &-1.875 \end{bmatrix}$

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:

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

= 0.3(2.3)- 1.7(0.5)

= 0.69 - 0.85

= -0.16

Therefore,

$A^{-1}=\frac{1}{-0.16} \begin{bmatrix} 2. 3 &- 1. 7 \\ -0.5 & 0.3 \end{bmatrix}$

$=\begin{bmatrix} \frac{1}{-0.16} \cdot 2. 3 & \frac{1}{-0.16} \cdot(- 1. 7) \\ \\ \frac{1}{-0.16} \cdot(-0.5) & \frac{1}{-0.16} \cdot0.3 \end{bmatrix}$

$=\begin{bmatrix} -14.375 &10.625 \\ 3.125 &-1.875 \end{bmatrix}$

Report an Error

Example Question #1 : Orthogonal Matrices

Determine if the following matrix is orthogonal or not.

$A=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$

Possible Answers:

is an orthogonal matrix

is not an orthogonal matrix

Correct answer:

is an orthogonal matrix

Explanation:

To determine if a matrix is orthogonal, we need to multiply the matrix by it's transpose, and see if we get the identity matrix.

$A=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$ , $A^T=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$

$A.A^T=\begin{bmatrix} (-1)(-1)&(0)(0) \\ (0)(0)& (1)(1) \end{bmatrix}=\begin{bmatrix} 1&0\\ 0& 1 \end{bmatrix}$

Since we get the identity matrix, then we know that is an orthogonal matrix.

Report an Error

Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}-0.60587&0.79556\\0.79556&0.60587\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}&\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}&\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}&\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.60587&0.79556\\0.79556&0.60587\end{bmatrix}= \\&\text{The matrix is orthogonal.}\end{align*}$

Report an Error

Example Question #342 : Linear Algebra

$\begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}0.68567&0.12975&-0.71626\\0.14807&0.93855&0.31176\\0.71269&-0.31982&0.62433\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}0.68567&0.12975&-0.71626\\0.14807&0.93855&0.31176\\0.71269&-0.31982&0.62433\end{bmatrix}=1\\&\text{The matrix is orthogonal.}\end{align*}$

Report an Error

Example Question #3 : Orthogonal Matrices

$\begin{align*}&\text{Is }A=\begin{bmatrix}-0.85616&0.46933\\0.46933&0.96236\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.85616&0.46933\\0.46933&0.96236\end{bmatrix}=-1.0442\\&\text{The matrix is not orthogonal.}\end{align*}$

Report an Error

Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}-0.57605&-0.81742\\-0.81742&0.57605\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.57605&-0.81742\\-0.81742&0.57605\end{bmatrix}=-1\\&\text{The matrix is orthogonal.}\end{align*}$

Report an Error

Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Is the matrix }A=\begin{bmatrix}-0.44125&-0.89739\\-0.89739&0.44125\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.44125&-0.89739\\-0.89739&0.44125\end{bmatrix}=-1\\&\text{The matrix is orthogonal.}\end{align*}$

Report an Error

Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Is the matrix }A=\begin{bmatrix}-0.62126&-0.0069882&0.77059\\-0.86406&-0.14727&-0.61555\\0.011558&-1.0415&0.043005\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.62126&-0.0069882&0.77059\\-0.86406&-0.14727&-0.61555\\0.011558&-1.0415&0.043005\end{bmatrix}=1.0968\\&\text{The matrix is not orthogonal.}\end{align*}$

Report an Error

Example Question #2 : Orthogonal Matrices

$\begin{align*}&\text{Is the matrix }A=\begin{bmatrix}0.012505&-0.9358&-0.11296\\-0.84627&0.011244&0.49169\\-0.3916&0.21287&-0.83217\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}0.012505&-0.9358&-0.11296\\-0.84627&0.011244&0.49169\\-0.3916&0.21287&-0.83217\end{bmatrix}=0.85765\\&\text{The matrix is not orthogonal.}\end{align*}$

Report an Error

Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Is }A=\begin{bmatrix}-0.55978&-0.80992&0.17514\\0.55713&-0.52432&-0.64396\\0.61339&-0.2629&0.74474\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.55978&-0.80992&0.17514\\0.55713&-0.52432&-0.64396\\0.61339&-0.2629&0.74474\end{bmatrix}=1\\&\text{The matrix is orthogonal.}\end{align*}$

Report an Error

View Linear Algebra Tutors

Majid
Certified Tutor

Semnan University - Iran, Bachelor of Engineering, Civil Engineering. BHRC - Iran, Master of Engineering, Civil Engineering.

View Linear Algebra Tutors

Muhammad
Certified Tutor

Quaid-E-Azam University, Master of Science, Applied Mathematics. Quaid-E-Azam University, Master of Philosophy, Applied Lingu...

View Linear Algebra Tutors

Lauren
Certified Tutor

Christopher Newport University, Bachelor of Science, Neuroscience. Christopher Newport University, Bachelor of Science, Psych...

All Linear Algebra Resources

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

Popular Subjects

Chemistry Tutors in New York City, French Tutors in Miami, Spanish Tutors in Chicago, LSAT Tutors in San Francisco-Bay Area, Calculus Tutors in Washington DC, ACT Tutors in Philadelphia, LSAT Tutors in Atlanta, Reading Tutors in San Diego, GMAT Tutors in Miami, English Tutors in New York City

Popular Courses & Classes

LSAT Courses & Classes in San Diego, Spanish Courses & Classes in Chicago, LSAT Courses & Classes in Miami, GRE Courses & Classes in Seattle, Spanish Courses & Classes in Houston, Spanish Courses & Classes in Phoenix, ACT Courses & Classes in Philadelphia, ACT Courses & Classes in Denver, GMAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Houston

Popular Test Prep

ISEE Test Prep in San Francisco-Bay Area, SAT Test Prep in Atlanta, GRE Test Prep in Dallas Fort Worth, GMAT Test Prep in Denver, SSAT Test Prep in Seattle, ACT Test Prep in Denver, GMAT Test Prep in San Diego, LSAT Test Prep in Los Angeles, GMAT Test Prep in Washington DC, ACT Test Prep in Miami

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 : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #341 : Linear Algebra

Example Question #1 : Orthogonal Matrices

Example Question #1 : Orthogonal Matrices

Example Question #342 : Linear Algebra

Example Question #3 : Orthogonal Matrices

Example Question #1 : Orthogonal Matrices

Example Question #1 : Orthogonal Matrices

Example Question #1 : Orthogonal Matrices

Example Question #2 : Orthogonal Matrices

Example Question #1 : Orthogonal Matrices

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: