Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : The Identity Matrix and Diagonal 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 #1 : The Identity Matrix And Diagonal Matrices

Which of the following matrices is a scalar multiple of the identity matrix?

$\begin{pmatrix} 1&2 \\ 3&4 \end{pmatrix}$ , $\begin{pmatrix} e&0 \\ 0&e \end{pmatrix}$ , $\begin{pmatrix} 0& 2\\ 2&0 \end{pmatrix}$

Possible Answers:

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

$\begin{pmatrix} 0& 2\\ 2&0 \end{pmatrix}$

$\begin{pmatrix} 1&2 \\ 3&4 \end{pmatrix}$

Correct answer:

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

Explanation:

The x identity matrix is

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

For this problem we see that

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

And so

$\begin{pmatrix} e&0 \\ 0&e \end{pmatrix}$ is a scalar multiple of the identity matrix.

Report an Error

Example Question #1 : The Identity Matrix And Diagonal Matrices

Which of the following is true concerning diagonal matrices?

Possible Answers:

The product of two diagonal matrices (in either order) is always another diagonal matrix.

The trace of any diagonal matrix is equal to its determinant.

The determinant of any diagonal matrix is .

The zero matrix (of any size) is not a diagonal matrix.

All of the other answers are false.

Correct answer:

The product of two diagonal matrices (in either order) is always another diagonal matrix.

Explanation:

You can verify this directly by proving it, or by multiplying a few examples on your calculator.

Report an Error

Example Question #1 : Operations And Properties

Which of the following is true concerning the $n \times n$ identity matrix I_n ?

Possible Answers:

Tr(I_n) = n

|I_n| = n

All of the other answers are true.

I_n^n = nI_n

Correct answer:

Tr(I_n) = n

Explanation:

is the trace operation. It means to add up the entries along the main diagonal of the matrix. Since I_n has ones along its main diagonal, the trace of I_n is .

Report an Error

Example Question #1 : The Identity Matrix And Diagonal Matrices

$A = \begin{bmatrix} 2& 0& 0& 0& 0\\ 0& 3& 0& 0& 0\\ 0& 0& 4& 0& 0\\ 0& 0& 0& 2& 0\\ 0& 0& 0& 0& 2 \end{bmatrix}$

Find A^7 .

Possible Answers:

$\begin{bmatrix} 128& 0& 0& 0& 0\\ 0& 2187& 0& 0& 0\\ 0& 0& 16384& 0& 0\\ 0& 0& 0& 128& 0\\ 0& 0& 0& 0& 128 \end{bmatrix}$

$\begin{bmatrix} 64& 0& 0& 0& 0\\ 0& 433& 0& 0& 0\\ 0& 0& 10& 0& 0\\ 0& 0& 0& 98& 0\\ 0& 0& 0& 0& 65 \end{bmatrix}$

None of the other answers

$\begin{bmatrix} 128& 0& 0& 0& 0\\ 0& 2187& 0& 0& 0\\ 0& 0& 16384& 0& 0\\ 0& 0& 0& 128& 0\\ 0& 0& 0& 0& 128 \end{bmatrix}$

$\begin{bmatrix} 986& 0& 0& 0& 0\\ 0& 2953& 0& 0& 0\\ 0& 0& 1638& 0& 0\\ 0& 0& 0& 900& 0\\ 0& 0& 0& 0& 900 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 128& 0& 0& 0& 0\\ 0& 2187& 0& 0& 0\\ 0& 0& 16384& 0& 0\\ 0& 0& 0& 128& 0\\ 0& 0& 0& 0& 128 \end{bmatrix}$

Explanation:

Since is a diagonal matrix, we can find it's powers more easily by raising the numbers inside it to the power in question.

$A^7 = \begin{bmatrix} 2^7& 0& 0& 0& 0\\ 0& 3^7& 0& 0& 0\\ 0& 0& 4^7& 0& 0\\ 0& 0& 0& 2^7& 0\\ 0& 0& 0& 0& 2^7 \end{bmatrix} = \begin{bmatrix} 128& 0& 0& 0& 0\\ 0& 2187& 0& 0& 0\\ 0& 0& 16384& 0& 0\\ 0& 0& 0& 128& 0\\ 0& 0& 0& 0& 128 \end{bmatrix}$

Report an Error

Example Question #1 : The Identity Matrix And Diagonal Matrices

True or false, the set of all $n \times n$ diagonal matrices forms a subspace of the vector space of all $n \times n$ matrices.

Possible Answers:

True

False

Correct answer:

True

Explanation:

To see why it's true, we have to check the two axioms for a subspace.

1. Closure under vector addition: is the sum of two diagonal matrices another diagonal matrix? Yes it is, only the diagonal entries are going to change, if at all. Nonetheless, it's still a diagonal matrix since all the other entries in the matrix are .

2. Closure under scalar multiplication: is a scalar times a diagonal matrix another diagonal matrix? Yes it is. If you multiply any number to a diagonal matrix, only the diagonal entries will change. All the other entries will still be .

Report an Error

Example Question #2 : The Identity Matrix And Diagonal Matrices

True or false, if any of the main diagonal entries of a diagonal matrix is , then that matrix is not invertible.

Possible Answers:

True

False

Correct answer:

True

Explanation:

Probably the simplest way to see this is true is to take the determinant of the diagonal matrix. We can take the determinant of a diagonal matrix by simply multiplying all of the entries along its main diagonal. Since one of these entries is , then the determinant is , and hence the matrix is not invertible.

Report an Error

Example Question #4 : The Identity Matrix And Diagonal Matrices

True or False, the $n \times n$ identity matrix has distinct (different) eigenvalues.

Possible Answers:

True

False

Correct answer:

False

Explanation:

We can find the eigenvalues of the identity matrix by finding all values of $\lambda$ such that $det(I_n\lambda -I_n) =0$ .

Hence we have

$det(I_n\lambda -I_n) = (\lambda-1)^n = 0$

So $\lambda =1$ is the only eigenvalue, regardless of the size of the identity matrix.

Report an Error

Example Question #5 : The Identity Matrix And Diagonal Matrices

What is the name for a matrix obtained by performing a single elementary row operation on the identity matrix?

Possible Answers:

An elementary matrix

An inverse matrix

A transition matrix

None of the other answers

An elementary row matrix

Correct answer:

An elementary matrix

Explanation:

This is the correct term. Elementary matrices themselves can be used in place of elementary row operations when row reducing other matrices when convenient.

Report an Error

Example Question #6 : The Identity Matrix And Diagonal Matrices

By definition, a square matrix that is similar to a diagonal matrix is

Possible Answers:

symmetric

idempotent

None of the given answers

diagonalizable

the identity matrix

Correct answer:

diagonalizable

Explanation:

Another way to state this definition is that a square matrix is said to diagonalizable if and only if there exists some invertible matrix and diagonal matrix such that $A=PBP^{-1}$ .

Report an Error

Example Question #7 : The Identity Matrix And Diagonal Matrices

The $n \times n$ identity matrix

Possible Answers:

has nullity .

has rank .

is idempotent.

is not diagonalizable.

has distinct eigenvalues, regardless of size.

Correct answer:

is idempotent.

Explanation:

An idempotent matrix is one such that A^2 =A . This is satisfied by the identity matrix since the identity matrix times itself is once again the identity matrix.

Report an Error

View Linear Algebra Tutors

Karen
Certified Tutor

Oklahoma Panhandle State University, Bachelor of Science, Mathematics. Oklahoma Panhandle State University, Bachelor in Arts,...

View Linear Algebra Tutors

Ping
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science. University of North Texas, Master of Arts, Education.

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

GRE Tutors in New York City, Physics Tutors in Atlanta, French Tutors in Los Angeles, Computer Science Tutors in San Diego, Chemistry Tutors in Los Angeles, Computer Science Tutors in Phoenix, Algebra Tutors in Miami, GMAT Tutors in Seattle, Math Tutors in Philadelphia, ISEE Tutors in Philadelphia

Popular Courses & Classes

MCAT Courses & Classes in San Diego, GMAT Courses & Classes in Miami, Spanish Courses & Classes in Atlanta, SAT Courses & Classes in Denver, LSAT Courses & Classes in Chicago, SAT Courses & Classes in Miami, SSAT Courses & Classes in Atlanta, Spanish Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Phoenix, GRE Courses & Classes in San Diego

Popular Test Prep

ACT Test Prep in New York City, MCAT Test Prep in Seattle, ISEE Test Prep in Phoenix, GMAT Test Prep in Los Angeles, LSAT Test Prep in Houston, SSAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in Houston, SAT Test Prep in Boston, SAT Test Prep in San Diego, GRE Test Prep in Atlanta

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 Identity Matrix and Diagonal Matrices

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #1 : The Identity Matrix And Diagonal Matrices

Example Question #1 : The Identity Matrix And Diagonal Matrices

Example Question #1 : Operations And Properties

Example Question #1 : The Identity Matrix And Diagonal Matrices

Example Question #1 : The Identity Matrix And Diagonal Matrices

Example Question #2 : The Identity Matrix And Diagonal Matrices

Example Question #4 : The Identity Matrix And Diagonal Matrices

Example Question #5 : The Identity Matrix And Diagonal Matrices

Example Question #6 : The Identity Matrix And Diagonal Matrices

Example Question #7 : The Identity Matrix And Diagonal Matrices

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: