Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : Operations and Properties

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 #111 : Operations And Properties

$\begin{align*}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}-3&-1&-16\\4&-11&-5\\3&-10&-9\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For the matrix }A=\begin{bmatrix}-3&-1&-16\\4&-11&-5\\3&-10&-9\end{bmatrix}\\&\text{We find: }\\&Tr(A)=(-3)+(-11)+(-9)=-23\end{align*}$

Report an Error

Example Question #112 : Operations And Properties

$\begin{align*}&\text{Calculate the trace of the matrix}\\&\begin{bmatrix}3&-16&17&16\\13&-10&4&-20\\-3&-8&-14&-13\\-3&-17&4&-1\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{The trace of a square nxn matrix is the sum of its diagonal elements:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{For the matrix }\begin{bmatrix}3&-16&17&16\\13&-10&4&-20\\-3&-8&-14&-13\\-3&-17&4&-1\end{bmatrix}\\&\text{This becomes: }\\&(3)+(-10)+(-14)+(-1)=-22\end{align*}$

Report an Error

Example Question #113 : Operations And Properties

$\begin{align*}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}1&6&-4&13\\9&19&1&-7\\-16&5&11&-3\\-17&-10&-14&-9\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For the matrix }A=\begin{bmatrix}1&6&-4&13\\9&19&1&-7\\-16&5&11&-3\\-17&-10&-14&-9\end{bmatrix}\\&\text{We find: }\\&Tr(A)=(1)+(19)+(11)+(-9)=22\end{align*}$

Report an Error

Example Question #14 : The Trace

$\begin{align*}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}6&19&-11&7&-9&7&8&-18\\-10&-11&7&14&-6&12&7&-20\\4&-5&17&-20&-2&-3&-2&11\\-7&12&-1&-19&-13&9&-1&-14\\-7&4&-13&10&-11&17&-9&11\\-13&-9&-17&3&8&2&-3&6\\6&7&6&18&-12&9&-11&-16\\4&-2&-2&7&11&-6&7&-3\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For the matrix }A=\begin{bmatrix}6&19&-11&7&-9&7&8&-18\\-10&-11&7&14&-6&12&7&-20\\4&-5&17&-20&-2&-3&-2&11\\-7&12&-1&-19&-13&9&-1&-14\\-7&4&-13&10&-11&17&-9&11\\-13&-9&-17&3&8&2&-3&6\\6&7&6&18&-12&9&-11&-16\\4&-2&-2&7&11&-6&7&-3\end{bmatrix}\\&\text{We find: }\\&Tr(A)=(6)+(-11)+(17)+(-19)+(-11)+(2)+(-11)+(-3)=-30\end{align*}$

Report an Error

Example Question #114 : Operations And Properties

$\begin{align*}&\text{Calculate the trace of the matrix}\\&\begin{bmatrix}15&-10&-7&-16&18\\6&-1&6&2&6\\2&9&1&20&-12\\-16&-16&-18&-4&-2\\-6&11&5&11&18\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{The trace of a square nxn matrix is the sum of its diagonal elements:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{For the matrix }\begin{bmatrix}15&-10&-7&-16&18\\6&-1&6&2&6\\2&9&1&20&-12\\-16&-16&-18&-4&-2\\-6&11&5&11&18\end{bmatrix}\\&\text{This becomes: }\\&(15)+(-1)+(1)+(-4)+(18)=29\end{align*}$

Report an Error

Example Question #114 : Operations And Properties

$\begin{align*}&\text{Compute }Tr\begin{bmatrix}1&15&-1&-4&7\\10&1&-6&-14&4\\-10&-19&10&-11&-2\\8&-6&10&-4&8\\8&-2&-20&-7&-3\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Tr in the context of linear algebra represents the trace.}\\&\text{We can find this quantity for a square nxn matrix by summing the elements of its diagonal:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{Doing so we find for}\begin{bmatrix}1&15&-1&-4&7\\10&1&-6&-14&4\\-10&-19&10&-11&-2\\8&-6&10&-4&8\\8&-2&-20&-7&-3\end{bmatrix}\\&\text{We find:}\\&(1)+(1)+(10)+(-4)+(-3)=5\end{align*}$

Report an Error

Example Question #114 : Operations And Properties

$\begin{align*}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}13&10&-5&-12\\12&18&-7&7\\-3&14&11&-14\\15&20&1&16\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For the matrix }A=\begin{bmatrix}13&10&-5&-12\\12&18&-7&7\\-3&14&11&-14\\15&20&1&16\end{bmatrix}\\&\text{We find: }\\&Tr(A)=(13)+(18)+(11)+(16)=58\end{align*}$

Report an Error

Example Question #16 : The Trace

$\begin{align*}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}12&-7&1&-17&-16&-15&7\\0&-13&0&-14&-18&14&2\\18&8&3&13&16&20&-20\\15&5&20&1&-1&12&-11\\0&16&3&14&10&4&-10\\7&-17&5&7&9&16&20\\11&3&18&3&-20&-16&15\end{bmatrix}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For the matrix }A=\begin{bmatrix}12&-7&1&-17&-16&-15&7\\0&-13&0&-14&-18&14&2\\18&8&3&13&16&20&-20\\15&5&20&1&-1&12&-11\\0&16&3&14&10&4&-10\\7&-17&5&7&9&16&20\\11&3&18&3&-20&-16&15\end{bmatrix}\\&\text{We find: }\\&Tr(A)=(12)+(-13)+(3)+(1)+(10)+(16)+(15)=44\end{align*}$

Report an Error

Example Question #193 : Linear Algebra

$A = \begin{bmatrix} 3&0 & 0\\ 0& 6& 0\\ 0&0 &x \end{bmatrix}$ ,

where is a complex number. The trace of $A^{2}$ is 50.

Evaluate .

Possible Answers:

$\pm i\sqrt{5}$

$\pm 5 i$

$\pm \sqrt{5}$

$\pm 5$

Correct answer:

$\pm \sqrt{5}$

Explanation:

is a diagonal matrix - its only nonzero elements are on the main diagonal, from upper left to lower right - so its square can be taken by simply squaring the diagonal elements. Since

$A = \begin{bmatrix} 3&0 & 0\\ 0& 6& 0\\ 0&0 &x \end{bmatrix}$

it follows that

$A ^{2}= \begin{bmatrix} 3^{2}&0 & 0\\ 0& 6^{2}& 0\\ 0&0 &x^{2} \end{bmatrix} = \begin{bmatrix} 9&0 & 0\\ 0& 36 & 0\\ 0&0 &x^{2} \end{bmatrix}$

The trace of a matrix is equal to the sum of the elements in its main diagonal, so the trace of $A^{2}$ is

$Tr(A^{2}) = 9 + 36 + x^{2}$

$Tr(A^{2}) = x^{2} + 45$

Since the trace is given to be 50, set this equal to 50 and solve for :

$x^{2} + 45= 50$

$x^{2} =5$

$x= \pm \sqrt{5}$

Report an Error

Example Question #15 : The Trace

$A = \begin{bmatrix} 3& -1& 2& a\\ 2& a& 4&-3 \\ 1& 4& a& 2\\ a& -4& 6& 3 \end{bmatrix}$

Evaluate so that the trace of is equal to 10.

Possible Answers:

The trace of cannot be 10 regardless of the value of

a = 3

a = 2

a = 1

a= 4

Correct answer:

a = 2

Explanation:

The trace of a matrix is equal to the sum of the elements in its main diagonal - the elements going from upper left to lower right. Therefore,

$\textrm{Tr} (A) = 3+a+a+3$

$\textrm{Tr} (A) = 2a+6$

Since the trace of is equal to 10, set 2a+6 equal to this and solve for :

2a+6 = 10

2a=4

a = 2

Report an Error

View Linear Algebra Tutors

Amanda
Certified Tutor

College of William and Mary, Bachelor in Arts, Anthropology.

View Linear Algebra Tutors

Alejandro
Certified Tutor

University of Western Ontario, Doctorate (PhD), Mathematics.

View Linear Algebra Tutors

Vandana
Certified Tutor

Jawaharlal Nehru Technological University, Bachelor of Technology, Civil Engineering. Florida International University, Maste...

All Linear Algebra Resources

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

Popular Subjects

Chemistry Tutors in Seattle, Algebra Tutors in Dallas Fort Worth, Computer Science Tutors in Philadelphia, Computer Science Tutors in New York City, SAT Tutors in San Francisco-Bay Area, Computer Science Tutors in Seattle, Spanish Tutors in Seattle, Biology Tutors in Miami, Algebra Tutors in Washington DC, ACT Tutors in Dallas Fort Worth

Popular Courses & Classes

Spanish Courses & Classes in Seattle, SSAT Courses & Classes in Chicago, ISEE Courses & Classes in San Diego, ACT Courses & Classes in Chicago, LSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Seattle, GRE Courses & Classes in Miami, ACT Courses & Classes in Philadelphia, GMAT Courses & Classes in Seattle, SSAT Courses & Classes in Atlanta

Popular Test Prep

MCAT Test Prep in Denver, SAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Denver, ISEE Test Prep in Philadelphia, GMAT Test Prep in Seattle, GMAT Test Prep in Washington DC, GMAT Test Prep in Phoenix, ISEE Test Prep in Denver, GRE Test Prep in Houston, ACT Test Prep in Seattle

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 : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #111 : Operations And Properties

Example Question #112 : Operations And Properties

Example Question #113 : Operations And Properties

Example Question #14 : The Trace

Example Question #114 : Operations And Properties

Example Question #114 : Operations And Properties

Example Question #114 : Operations And Properties

Example Question #16 : The Trace

Example Question #193 : Linear Algebra

Example Question #15 : The Trace

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: