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

← Previous 1 2 … 6 7 8 9 10 11 12 13 14 … 91 92 Next →

Example Question #12 : The Identity Matrix And Diagonal Matrices

True or false:

$\begin{bmatrix} 2&0 & 0& 0 & 0\\ 0&-7 &0 &0 &0 \\ 0& 0& 5&0 & 0 \\ 0& 0& 0& -4 & 0\\ 0& 0& 0& 0 &0 \end{bmatrix}$

is an example of a diagonal matrix.

Possible Answers:

False

True

Correct answer:

True

Explanation:

A matrix is diagonal if and only if $a_{m, n} = 0$ - that is, the element in column , row is equal to zero - for all $m \ne n$ . The given matrix fits this criterion.

Report an Error

Example Question #13 : The Identity Matrix And Diagonal Matrices

True or false:

$\begin{bmatrix} -0.1& 0 &0 &0 &0 &0 \\ 0&0.2 &0 &0 &0 &0 \\ 0& 0& -0.3&0 &0 &0 \\ 0& 0& 0& 0.4& 0& 0\\ 0& 0& 0& 0& -0.5& 0\\ 0 &0 &0 &0 &0 & 0.6 \end{bmatrix}$

is an example of a diagonal matrix.

Possible Answers:

False

True

Correct answer:

True

Explanation:

A matrix is diagonal if and only if $a_{m, n} = 0$ - that is, the element in column , row is equal to zero - for all $m \ne n$ . The given matrix satisfies this condition, since its only nonzero elements are the first element in Column 1, the second element in Column 2, and so forth.

Report an Error

Example Question #15 : The Identity Matrix And Diagonal Matrices

$A = \begin{bmatrix}1 &0 &0 &0 &0 \\ 0& 0.3 & 0& 0& 0\\ 0&0 &0.5 & 0& 0\\ 0& 0& 0& 0.7& 0\\ 0& 0& 0& 0& 0.9 \end{bmatrix}$

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

Possible Answers:

$A^{-1} = \begin{bmatrix}1 &0 &0 &0 &0 \\ 0& \frac{10}{3} & 0& 0& 0\\ 0&0 &2 & 0& 0\\ 0& 0& 0& \frac{10}{7}& 0\\ 0& 0& 0&0& \frac{10}{9} \end{bmatrix}$

$A ^{-1}= \begin{bmatrix} 0 &0 &0 &0 &1 \\ 0 & 0& 0& \frac{10}{3} & 0\\ 0&0 & 2& 0 & 0\\ 0& \frac{10}{7}& 0& 0&0\\ \frac{10}{9} & 0& 0& 0&0 \end{bmatrix}$

$A^{-1} = \begin{bmatrix}-1 &0 &0 &0 &0 \\ 0&- 0.3 & 0& 0& 0\\ 0&0 &-0.5 & 0& 0\\ 0& 0& 0& -0.7& 0\\ 0& 0& 0& 0&- 0.9 \end{bmatrix}$

$A ^{-1}= \begin{bmatrix}0 &0 &0 &0 &-1 \\ 0& 0& 0& - 0.3& 0\\ 0&0 &-0.5 & 0& 0\\ 0& -0.7& 0&0& 0\\ - 0.9& 0& 0& 0&0 \end{bmatrix}$

$A ^{-1}$ is undefined.

Correct answer:

$A^{-1} = \begin{bmatrix}1 &0 &0 &0 &0 \\ 0& \frac{10}{3} & 0& 0& 0\\ 0&0 &2 & 0& 0\\ 0& 0& 0& \frac{10}{7}& 0\\ 0& 0& 0&0& \frac{10}{9} \end{bmatrix}$

Explanation:

is a diagonal matrix - its only nonzero entries are in its main diagonal, which comprises the elements in row , column , for m = n .

The inverse $A^{-1}$ of a such a matrix can be found simply by replacing each element in the main diagonal with its reciprocal. Rewriting the elements in the diagonal matrix as fractions,

$A = \begin{bmatrix}1 &0 &0 &0 &0 \\ 0& \frac{3}{10} & 0& 0& 0\\ 0&0 & \frac{5}{10} & 0& 0\\ 0& 0& 0& \frac{7}{10}& 0\\ 0& 0& 0&0& \frac{9}{10} \end{bmatrix}$ ,

$A = \begin{bmatrix}1 &0 &0 &0 &0 \\ 0& \frac{3}{10} & 0& 0& 0\\ 0&0 & \frac{1}{2} & 0& 0\\ 0& 0& 0& \frac{7}{10}& 0\\ 0& 0& 0&0& \frac{9}{10} \end{bmatrix}$ .

Replace each diagonal element with its reciprocal to find $A^{-1}$ :

$A^{-1} = \begin{bmatrix}1 &0 &0 &0 &0 \\ 0& \frac{10}{3} & 0& 0& 0\\ 0&0 &2 & 0& 0\\ 0& 0& 0& \frac{10}{7}& 0\\ 0& 0& 0&0& \frac{10}{9} \end{bmatrix}$

Report an Error

Example Question #91 : Linear Algebra

$A = \begin{bmatrix} e & 1+ i & 1 \\ 0 & \pi & 1 - i \\ 0 & 0 & i \end{bmatrix}$

True or false: is an example of a diagonal matrix.

Possible Answers:

True

False

Correct answer:

False

Explanation:

A diagonal matrix has only zeroes as entries off of its main (upper-left to lower-right) diagonal. has three nonzero entries off this diagonal (Row 1, column 2, for example), so it is not a diagonal matrix.

Report an Error

Example Question #11 : Operations And Properties

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

is a strictly diagonally dominant matrix for what range of values of ?

Possible Answers:

|a| > 5

|a| > 1

|a| > 2

|a| > 3

|a| > 4

Correct answer:

|a| > 4

Explanation:

An $n \times n$ matrix is strictly diagonally dominant if, for each $j \in 1, 2, ...n$ , it holds that on Row , the absolute value of the element in Column is greater than the sum of the absolute values of the other elements in that row. Therefore, for to be strictly diagonally dominant, the following must hold:

For Row 1: |a| > |1| + |-2|

or |a| > 3

For Row 2: |a| > |-3| + |1|

or |a| > 4

For Row 3: |a| > |1| + |2| ,

or |a| >3

For all three conditions to hold, it is necessary that |a| > 4 . This is the correct choice.

Report an Error

Example Question #11 : The Identity Matrix And Diagonal Matrices

is a diagonal matrix such that $A^{100} = 2 I$ , where refers to the $2 \times 2$ identity.

can be one of how many matrices?

Possible Answers:

10,000

$2^{100}$

Correct answer:

10,000

Explanation:

is a diagonal matrix, and its dimensions are $2 \times 2$ , so, for some complex and (the problem did not specify that the entries were real),

$A = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$ .

To raise a diagonal matrix to a power, simply raise each nonzero element to that power. It holds that

$A^{100} = \begin{bmatrix} a ^{100} & 0 \\ 0 & b^{100} \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$ ,

and, consequently, $a^{100} = b^{100} = 2$ .

Equivalently, both and must be a one-hundredth root of 2, of which there are 100. Therefore, the number of possible matrices for is $100 \times 100 = 10,000$ .

Report an Error

Example Question #11 : Operations And Properties

What is the trace of the $2 \times 3$ identity matrix?

Possible Answers:

None of the other answers

Correct answer:

None of the other answers

Explanation:

This question does not make sense since there is no such thing as the $2 \times 3$ identity matrix, and it is not possible to take the trace of a matrix that is not square. This question is mostly meant to test your ability to think critically when reading certain mathematics problems.

Report an Error

Example Question #92 : Linear Algebra

True or false; all diagonal matrices are diagonalizable.

Possible Answers:

False

True

Correct answer:

True

Explanation:

A matrix is diagonalizable if it can be written as $A = P^{-1}DA$ , where is any invertible matrix, and is any diagonal matrix. If is already a diagonal matrix, we can of course write it as $I^{-1}AI$ . Hence any diagonal matrix is diagonalizable.

Report an Error

Example Question #91 : Linear Algebra

Which of the following is an identity matrix?

Possible Answers:

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

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

All of these are valid identitiy matricies.

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

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

Correct answer:

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

Explanation:

An identity matrix is a square matrix in which all diagonal elements are 1 and all non-diagonal elements are 0. The only square matricies above are:

$\begin{bmatrix} 1& 0& 0\\ 0& 1& 0\\ 0&0 &1 \end{bmatrix} \textup{and}$ $\begin{bmatrix} 0& 0&1 \\ 0& 1 & 0\\ 1& 0&0 \end{bmatrix}$

The former of these two matricies is the only one fitting the critera of diagonal elements (diagonal meaning from top left to bottom right) being 1 and non-diagonal elements being 0.

Report an Error

Example Question #21 : The Identity Matrix And Diagonal Matrices

Which of the following is a diagonal matrix?

Possible Answers:

$\begin{bmatrix} 1&0 &2 \\ 0& 3& 0\\ 4& 0& 5 \end{bmatrix}$

$\begin{bmatrix} 1& 0& 0\\ 0& 2& 0\\ 0&0 & 3 \end{bmatrix}$

$\begin{bmatrix} 1& 2&0 &0 \\ 0& 0& 3& 4 \end{bmatrix}$

$\begin{bmatrix} 0& 1& 2\\ 3& 0& 4\\ 5&6 & 0 \end{bmatrix}$

$\begin{bmatrix} 0& 0& 1\\ 0& 2& 0\\ 3&0 & 0 \end{bmatrix}\,$

Correct answer:

$\begin{bmatrix} 1& 0& 0\\ 0& 2& 0\\ 0&0 & 3 \end{bmatrix}$

Explanation:

A diagonal matrix is a matrix in which all elements except diagonal elements are zero. A diagonal element of a matrix M is the element M_ij for which i=j. Here, i refers to the number of columns and j is the number of rows. The only matrix that fufills this definition is:

$\begin{bmatrix} 1& 0& 0\\ 0& 2& 0\\ 0&0 & 3 \end{bmatrix}$

Report an Error

← Previous 1 2 … 6 7 8 9 10 11 12 13 14 … 91 92 Next →

View Linear Algebra Tutors

Yoonsik
Certified Tutor

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

View Linear Algebra Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Linear Algebra Tutors

Godswill
Certified Tutor

Rivers State University, Bachelor of Engineering, Mechanical Engineering. Texas Southern University, Master of Science, Compu...

All Linear Algebra Resources

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

Popular Subjects

French Tutors in Denver, English Tutors in San Francisco-Bay Area, Spanish Tutors in New York City, Reading Tutors in Miami, GMAT Tutors in Houston, GMAT Tutors in Miami, Calculus Tutors in Philadelphia, ACT Tutors in Denver, Chemistry Tutors in Washington DC, Calculus Tutors in Dallas Fort Worth

Popular Courses & Classes

MCAT Courses & Classes in Seattle, LSAT Courses & Classes in Seattle, GMAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Denver, SAT Courses & Classes in Phoenix, Spanish Courses & Classes in Miami, GMAT Courses & Classes in Atlanta, Spanish Courses & Classes in Washington DC, SSAT Courses & Classes in San Diego, Spanish Courses & Classes in Dallas Fort Worth

Popular Test Prep

MCAT Test Prep in Houston, SAT Test Prep in San Francisco-Bay Area, ACT Test Prep in Philadelphia, ISEE Test Prep in Philadelphia, GRE Test Prep in San Diego, SSAT Test Prep in Washington DC, GRE Test Prep in Philadelphia, SAT Test Prep in Chicago, ACT Test Prep in San Francisco-Bay Area, 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 : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #12 : The Identity Matrix And Diagonal Matrices

Example Question #13 : The Identity Matrix And Diagonal Matrices

Example Question #15 : The Identity Matrix And Diagonal Matrices

Example Question #91 : Linear Algebra

Example Question #11 : Operations And Properties

Example Question #11 : The Identity Matrix And Diagonal Matrices

Example Question #11 : Operations And Properties

Example Question #92 : Linear Algebra

Example Question #91 : Linear Algebra

Example Question #21 : 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: