Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : Orthogonal Matrices

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 #11 : Orthogonal Matrices

$\begin{align*}&\text{Is }A=\begin{bmatrix}-0.51053&0.15898&-0.84503\\-0.61849&-0.75062&0.23245\\-0.59735&0.64132&0.48155\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

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:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.51053&0.15898&-0.84503\\-0.61849&-0.75062&0.23245\\-0.59735&0.64132&0.48155\end{bmatrix}=1\\&\text{The matrix is orthogonal.}\end{align*}$

Report an Error

Example Question #12 : Orthogonal Matrices

$\begin{align*}&\text{Determine if the matrix }A=\begin{bmatrix}-0.56605&-0.7545&-0.33214\\0.76145&-0.32415&-0.56136\\0.31588&-0.57066&0.758\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

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:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.56605&-0.7545&-0.33214\\0.76145&-0.32415&-0.56136\\0.31588&-0.57066&0.758\end{bmatrix}=1\\&\text{The matrix is orthogonal.}\end{align*}$

Report an Error

Example Question #13 : Orthogonal Matrices

$\begin{align*}&\text{Is the matrix }A=\begin{bmatrix}-0.58762&-0.34601&-1.5258\\0.13555&-1.2568&-0.72109\\-1.2767&-1.2221&-0.65535\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

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:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.58762&-0.34601&-1.5258\\0.13555&-1.2568&-0.72109\\-1.2767&-1.2221&-0.65535\end{bmatrix}=2.3855\\&\text{The matrix is not orthogonal.}\end{align*}$

Report an Error

Example Question #14 : Orthogonal Matrices

Which of the matrices is orthogonal?

$\begin{pmatrix} 0&0 \\ 0&0 \end{pmatrix}$ , $\begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}$ , $\begin{pmatrix} 1&1 \\ 0&0 \end{pmatrix}$ , $\begin{pmatrix} 0&0 \\ 1&1 \end{pmatrix}$

Possible Answers:

$\begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}$

$\begin{pmatrix} 1&1 \\ 0&0 \end{pmatrix}$

$\begin{pmatrix} 0&0 \\ 1&1 \end{pmatrix}$

$\begin{pmatrix} 0&0 \\ 0&0 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}$

Explanation:

An x matrix is defined to be orthogonal if

AA^t=I

where is the x identity matrix.

We see that

$\begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}$ $\begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}=\begin{pmatrix} 1*1+0*0&1*0+0*-1 \\ 0*1+(-1)*0&0*0+(-1)*(-1) \end{pmatrix}$

$=\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}$

And so

$\begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}$ is orthogonal.

Report an Error

Example Question #15 : Orthogonal Matrices

Which of the matrices is orthogonal?

$\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}$ , $\begin{pmatrix} 1&1 \\ 1&1 \end{pmatrix}$ , $\begin{pmatrix} 1&1 \\ 0&0 \end{pmatrix}$ , $\begin{pmatrix} 0&0 \\ 1&1 \end{pmatrix}$

Possible Answers:

$\begin{pmatrix} 1&1 \\ 0&0 \end{pmatrix}$

$\begin{pmatrix} 0&0 \\ 1&1 \end{pmatrix}$

$\begin{pmatrix} 1&1 \\ 1&1 \end{pmatrix}$

$\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}$

Explanation:

An x matrix is defined to be orthogonal if

AA^t=I

where is the x identity matrix.

We see that

$\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}$ $\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}=\begin{pmatrix} 1*1+0*0&1*0+0*1 \\ 0*1+1*0&0*0+1*1 \end{pmatrix}$

$=\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}$

And so

$\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}$ is orthogonal.

Report an Error

Example Question #14 : Orthogonal Matrices

By definition, an orthogonal matrix is a square matrix such that

Possible Answers:

A=A^T

A^2 = I

A^n=0 for some positive integer

AA^T = I

-A=A^T

Correct answer:

AA^T = I

Explanation:

Notice that this also means that the transpose of an orthogonal matrix is its inverse.

Report an Error

Example Question #15 : Orthogonal Matrices

Assume M is an orthogonal matrix. Which of the following is not always true?

Possible Answers:

$\textbf{M}^T=\textbf{M}^{-1}$

$\textbf{M}^T\textbf{M}=\textbf{I}$

All of these options are always true.

$\textbf{M}=\textbf{M}^T$

$(\textup{det}(\textbf{M}))^2=1$

Correct answer:

$\textbf{M}=\textbf{M}^T$

Explanation:

Let us examine each of the options:

$\textbf{M}^T\textbf{M}=\textbf{I}$ This is the definition of an orthogonal matrix; it is always true.

$\textbf{M}^T=\textbf{M}^{-1}$ This can be directly proved from the previous statment. If you subtitute the inverse for the transpose in the definition equation, it is still true.

$(\textup{det}(\textbf{M}))^2=1$ The determinant of any orthogonal matrix is either 1 or -1. This statment can be proved in the following way:

$1=\textup{det}(\textbf{I})=\textup{det}(\textbf{M}^T \textbf{M}) =\textup{det}(\textbf{M}^T)\textup{det}(\textbf{M})=(\textup{det}(\textbf{M}))^2$

The incorrect statment is $\textbf{M}=\textbf{M}^T$ . Consider an example matrix:

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

which has a transpose

$\textbf{M}^T=\begin{bmatrix} 0& 1\\ -1&0 \end{bmatrix}$

M and its transpose are clearly not equal. However, if we multiply them, we can see that their product is the identity matrix and they are therefore orthogonal.

$\textbf{M}\textbf{M}^T=\begin{bmatrix} 0& -1\\ 1&0 \end{bmatrix} \begin{bmatrix} 0& 1\\ -1&0 \end{bmatrix}=\begin{bmatrix} 1&0 \\ 0& 1 \end{bmatrix} =\textbf{I}$

Report an Error

Example Question #16 : Orthogonal Matrices

The matrix M given below is orthogonal. What is x?

$\textbf{M}=\begin{bmatrix} x& -1\\ 1&0 \end{bmatrix}$

Possible Answers:

There is not enough information to determine x.

Correct answer:

Explanation:

We know that for any orthogonal matrix:

$\textbf{M}\textbf{M}^T=\textbf{I}$

So, we can set up an equation with our matrix. First, let's find the transpose of M:

$\textbf{M}=\begin{bmatrix} x& -1\\ 1&0 \end{bmatrix} \rightarrow\textbf{M}^T =\begin{bmatrix} x& 1\\ -1& 0 \end{bmatrix}$

Now, let's set up the equation based on the definition:

$\textbf{MM}^T=\begin{bmatrix} x& -1\\ 1&0 \end{bmatrix} \begin{bmatrix} x& 1\\ -1& 0 \end{bmatrix} =\begin{bmatrix} x^2+1& x\\ x&1 \end{bmatrix}=\begin{bmatrix} 1&0 \\ 0& 1 \end{bmatrix}=\textbf{I}$

Comparing the last two matricies, one can see that x=0.

Report an Error

Example Question #282 : Operations And Properties

The matrix A is given below. Is it orthogonal?

$\textbf{A}=\begin{bmatrix} 2& 1&0\\ -1& 3&1 \\ 1&0 &-1 \end{bmatrix}$

Possible Answers:

There is not enough intformation to determine whether or not A is orthogonal.

Yes, A is orthogonal.

No, A is not orthogonal.

Correct answer:

No, A is not orthogonal.

Explanation:

For a matrix M to be orthogonal, it has to satisfy the following condition:

$\textbf{MM}^T=\textbf{I}$

We can find the transpose of A and multiply it by A to determine whether or not it is orthogonal:

$\textbf{A}=\begin{bmatrix} 2& 1&0\\ -1& 3&1 \\ 1&0 &-1 \end{bmatrix} \rightarrow \textbf{A}^T= \begin{bmatrix} 2& -1&1\\ 1& 3&0 \\ 0&1&-1 \end{bmatrix}$

$\textbf{AA}^T= \begin{bmatrix} 2& 1&0\\ -1& 3&1 \\ 1&0 &-1 \end{bmatrix} \begin{bmatrix} 2& -1&1\\ 1& 3&0 \\ 0&1&-1 \end{bmatrix}= \begin{bmatrix} 5& 5&-2 \\ 5&11 &-2 \\ -2& -2& 2 \end{bmatrix} \neq \begin{bmatrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0&1 \end{bmatrix}$

Therefore, A is not an orthogonal matrix.

Report an Error

Example Question #16 : Orthogonal Matrices

The matrix B is given below. Is B orthogonal? (Round to three decimal places)

$\textbf{B}=\begin{bmatrix} 0.5257&-0.8507 \\ 0.8507& 0.5257 \end{bmatrix}$

Possible Answers:

No, B is not orthogonal.

There is not enough information to determine.

Yes, B is orthogonal.

Correct answer:

Yes, B is orthogonal.

Explanation:

For a matrix M to be orthogonal, it has to satisfy the following condition:

$\textbf{MM}^T=\textbf{I}$

We can find the transpose of B and multiply it by B to determine whether or not it is orthogonal:

$\textbf{B}=\begin{bmatrix} 0.5257&-0.8507 \\ 0.8507& 0.5257 \end{bmatrix}\rightarrow \textbf{B}^T=\begin{bmatrix} 0.5257&0.8507 \\ -0.8507& 0.5257 \end{bmatrix}$

$\textbf{B} \textbf{B}^T=\begin{bmatrix} 0.5257&-0.8507 \\ 0.8507& 0.5257 \end{bmatrix}\begin{bmatrix} 0.5257&0.8507 \\ -0.8507& 0.5257 \end{bmatrix}= \begin{bmatrix} 1.000&0 \\ 0& 1.000 \end{bmatrix}=\textbf{I}$

Report an Error

View Linear Algebra Tutors

Michael
Certified Tutor

Cleveland State University, Bachelor of Science, Mathematics.

View Linear Algebra Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

View Linear Algebra Tutors

Allison
Certified Tutor

Asbury University, Bachelor of Education, Mathematics Teacher Education.

All Linear Algebra Resources

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

Popular Subjects

Physics Tutors in Washington DC, Computer Science Tutors in Washington DC, LSAT Tutors in Miami, English Tutors in Denver, ISEE Tutors in Miami, ACT Tutors in Seattle, French Tutors in San Diego, Spanish Tutors in Seattle, French Tutors in Phoenix, ISEE Tutors in New York City

Popular Courses & Classes

Spanish Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Boston, GRE Courses & Classes in San Diego, ACT Courses & Classes in Los Angeles, SAT Courses & Classes in San Diego, ACT Courses & Classes in Miami, SSAT Courses & Classes in Washington DC, ISEE Courses & Classes in Houston, SAT Courses & Classes in Atlanta, SSAT Courses & Classes in Miami

Popular Test Prep

SSAT Test Prep in Denver, ACT Test Prep in Miami, SAT Test Prep in Denver, GRE Test Prep in Seattle, GMAT Test Prep in San Diego, GMAT Test Prep in Los Angeles, GMAT Test Prep in Denver, ISEE Test Prep in Phoenix, ISEE Test Prep in Houston, SSAT Test Prep in Los Angeles

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 : Orthogonal Matrices

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #11 : Orthogonal Matrices

Example Question #12 : Orthogonal Matrices

Example Question #13 : Orthogonal Matrices

Example Question #14 : Orthogonal Matrices

Example Question #15 : Orthogonal Matrices

Example Question #14 : Orthogonal Matrices

Example Question #15 : Orthogonal Matrices

Example Question #16 : Orthogonal Matrices

Example Question #282 : Operations And Properties

Example Question #16 : Orthogonal Matrices

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: