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 #21 : The Transpose

and $A ^{T}$ are both lower triangular square matrices. Which of the following must follow from this?

Possible Answers:

is a zero matrix

is an identity matrix

is a nonsingular matrix, but not necessarily the identity matrix.

is a singular matrix, but not necessarily a zero matrix.

is a diagonal matrix, but not necessarily an identity matrix or a zero matrix.

Correct answer:

is a diagonal matrix, but not necessarily an identity matrix or a zero matrix.

Explanation:

Let be a three-by-three matrix; this reasoning extends to square matrices of all sizes.

is a lower triangular matrix, so all of the entries above its main (upper left to lower right) diagonal are zeroes; that is,

$A = \begin{bmatrix} a&0 &0 \\ b& c& 0\\ d& e& f \end{bmatrix}$ .

$A ^{T}$ , the transpose of , is the matrix formed by switching rows with columns, so

$A^{T} = \begin{bmatrix} a&b&d \\ 0& c& e\\ 0& 0& f \end{bmatrix}$ .

However, $A ^{T}$ is lower triangular also; as a consequence,

b,d,e =0 ,

and

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

This demonstrates that must be a diagonal matrix - one with only zeroes off its main diagonal.

Report an Error

Example Question #51 : Operations And Properties

Let , , and be real numbers such that

$A = \begin{bmatrix} a& b& c\\ 1& 2& 3\\ -4&-5 &-6 \end{bmatrix}$ , $B = \begin{bmatrix} a&1 &-4 \\ b& 2& -5\\ c& 3& -6 \end{bmatrix}$ ,

and the determinant of is 8.

True or false: The determinant of is 8.

Possible Answers:

False

True

Correct answer:

True

Explanation:

is the transpose of - the matrix formed by interchanging the rows of with its columns. The determinant of a matrix and that of its transpose are equal, so, since has determinant 8, so does .

Report an Error

Example Question #52 : Operations And Properties

$A = \begin{bmatrix} 1 + i & 1 - i \\ 2 & -2 \end{bmatrix}$

Find $A^{*}$ .

Possible Answers:

$A^{*} = \begin{bmatrix} 1+ i & -2\\ 1 + i & 2 \end{bmatrix}$

$A^{*} = \begin{bmatrix} 1+ i & 2\\ 1 - i & -2 \end{bmatrix}$

$A^{*} = \begin{bmatrix} 1+ i & 2\\ 1 + i & -2 \end{bmatrix}$

$A^{*} = \begin{bmatrix} 1- i & 2\\ 1 + i & -2 \end{bmatrix}$

$A^{*} = \begin{bmatrix} 1- i & -2\\ 1 + i &2 \end{bmatrix}$

Correct answer:

$A^{*} = \begin{bmatrix} 1- i & 2\\ 1 + i & -2 \end{bmatrix}$

Explanation:

$A^{*}$ , the conjugate transpose of , is the result of transposing the matrix - interchanging rows with columns - and changing each entry to its complex conjugate. First, find transpose $A^{T}$ :

$A = \begin{bmatrix} 1 + i & 1 - i \\ 2 & -2 \end{bmatrix}$ ,

$A^{T} = \begin{bmatrix} 1 + i & 2\\ 1 - i & -2 \end{bmatrix}$

Change each entry to its complex conjugate:

$A^{*} = \begin{bmatrix} 1- i & 2\\ 1 + i & -2 \end{bmatrix}$ .

Report an Error

Example Question #53 : Operations And Properties

$A = \begin{bmatrix} 1 + i & 1 - i \\ 2 & -2 \end{bmatrix}$

Find $A^{T}$ .

Possible Answers:

$A^{T} = \begin{bmatrix} 1 + i & 2\\ 1 - i & -2 \end{bmatrix}$

None of the other choices gives the correct response.

$A^{T} = \begin{bmatrix} 1 + i & -2\\ 1 - i &2 \end{bmatrix}$

$A^{T} = \begin{bmatrix} 1 - i & 2\\ 1 + i & -2 \end{bmatrix}$

$A^{T} = \begin{bmatrix} 1 - i & -2\\ 1 + i & 2 \end{bmatrix}$

Correct answer:

$A^{T} = \begin{bmatrix} 1 + i & 2\\ 1 - i & -2 \end{bmatrix}$

Explanation:

$A^{T}$ , the transpose, is the result of switching the rows of with the columns.

$A = \begin{bmatrix} 1 + i & 1 - i \\ 2 & -2 \end{bmatrix}$ ,

$A^{T} = \begin{bmatrix} 1 + i & 2\\ 1 - i & -2 \end{bmatrix}$ .

Report an Error

Example Question #51 : Operations And Properties

and are skew-symmetric matrices.

Which of the following is true of $A^{T} B^{T}$ ?

Possible Answers:

$A^{T} B^{T} =- AB$

$A^{T} B^{T} = AB$

Correct answer:

$A^{T} B^{T} = AB$

Explanation:

By definition, the transpose $A^{T}$ of a skew-symmetric matrix is equal to its additive inverse . It follows that

$A^{T} B^{T} =( -A)(-B) =( -1) A (-1)B = (-1)(-1)AB = AB$

Report an Error

Example Question #55 : Operations And Properties

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

Find $A^{*}$ .

Possible Answers:

$\begin{bmatrix} 1 & -1 &0 \\ 1- i & 1 + i & -1 \\ 0 & 0 &- i \end{bmatrix}$

$\begin{bmatrix} 1 & 1 &0 \\ 1+ i & 1 - i & 1 \\ 0 & 0 & i \end{bmatrix}$

$\begin{bmatrix} 1 & 1 &0 \\ 1- i & 1 + i & 1 \\ 0 & 0 &- i \end{bmatrix}$

$\begin{bmatrix} 1 & 1 &0 \\ 1+ i & 1 - i & 1 \\ 0 & 0 &- i \end{bmatrix}$

$\begin{bmatrix} 1 & 1 &0 \\ 1-i & 1 +i & 1 \\ 0 & 0 & i \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1 & 1 &0 \\ 1- i & 1 + i & 1 \\ 0 & 0 &- i \end{bmatrix}$

Explanation:

$A^{*}$ , the conjugate transpose of , is the result of transposing the matrix - interchanging rows with columns - and changing each entry to its complex conjugate. First, find transpose $A^{T}$ :

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

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

Change each entry to its complex conjugate:

$A^{*}= \begin{bmatrix} 1 & 1 &0 \\ 1- i & 1 + i & 1 \\ 0 & 0 &- i \end{bmatrix}$

Report an Error

Example Question #56 : Operations And Properties

$A = \begin{bmatrix} -2 & 1 & 3 \\ 5 & 4 &- 1 \\ 4 & -2 & 1 \end{bmatrix}$

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

Possible Answers:

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

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

None of the other responses gives the correct answer.

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

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

Correct answer:

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

Explanation:

$A^{*}$ , the conjugate transpose of , is the result of transposing the matrix - interchanging rows with columns - and changing each entry to its complex conjugate. First, find transpose $A^{T}$ :

$A = \begin{bmatrix} -2 & 1 & 3 \\ 5 & 4 &- 1 \\ 4 & -2 & 1 \end{bmatrix}$

$A^{T}= \begin{bmatrix} -2 & 5& 4 \\ 1 & 4 &- 2 \\ 3& -1&1 \end{bmatrix}$

Each entry of $A^{*}$ is equal to the complex conjugate of the corresponding entry of $A^{T}$ . However, each entry in $A^{T}$ is real, so each entry is equal to its own complex conjugate, and

$A ^{*} = A^{T}= \begin{bmatrix} -2 & 5& 4 \\ 1 & 4 &- 2 \\ 3& -1&1 \end{bmatrix}$

Report an Error

Example Question #56 : Operations And Properties

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

Find $A^{T}$ .

Possible Answers:

$\begin{bmatrix} 1 & 1 &0 \\ 1+ i & 1 - i & 1 \\ 0 & 0 &- i \end{bmatrix}$

$\begin{bmatrix} 1 & 1 &0 \\ 1+ i & 1 - i & 1 \\ 0 & 0 & i \end{bmatrix}$

$\begin{bmatrix} 1 & 1 &0 \\ 1+i & 1 -i & 1 \\ 0 & 0 & i \end{bmatrix}$

None of the other choices gives the correct response.

$\begin{bmatrix} 1 & 1 &0 \\ 1-i & 1 +i & 1 \\ 0 & 0 & i \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1 & 1 &0 \\ 1+ i & 1 - i & 1 \\ 0 & 0 & i \end{bmatrix}$

Explanation:

$A^{T}$ , the transpose, is the result of switching the rows of with the columns.

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

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

Report an Error

Example Question #31 : The Transpose

$A^{*} = \begin{bmatrix} 3 + i & 5 - 2i \\3 + 2i & 3i \end{bmatrix}$

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

Possible Answers:

$A^{T} = \begin{bmatrix} 3 + i &3 - 2i \\ 5 + 2i & 3i \end{bmatrix}$

$A^{T} = \begin{bmatrix} 3 - i & 5 + 2i \\3 - 2i & 3i \end{bmatrix}$

$A^{T} = \begin{bmatrix} 3 - i & 5 + 2i \\3 - 2i & -3i \end{bmatrix}$

$A^{T} = \begin{bmatrix} 3 + i & 5 + 2i \\3 - 2i & 3i \end{bmatrix}$

$A^{T} = \begin{bmatrix} 3 - i &3 - 2i \\ 5 + 2i & -3i \end{bmatrix}$

Correct answer:

$A^{T} = \begin{bmatrix} 3 - i & 5 + 2i \\3 - 2i & -3i \end{bmatrix}$

Explanation:

$A^{T}$ is the transpose of - the result of interchanging the rows of with its columns. $A^{*}$ is the conjugate transpose of - the result of changing each entry of $A^{T}$ to its complex conjugate. Therefore, if

$A^{*} = \begin{bmatrix} 3 + i & 5 - 2i \\3 + 2i & 3i \end{bmatrix}$ ,

we can find $A^{T}$ by simply changing each entry in $A^{*}$ to its complex conjugate:

$A^{T} = \begin{bmatrix} \overline{3 + i }& \overline{5 - 2i} \\\overline{3 + 2i} & \overline{3i }\end{bmatrix}$

$A^{T} = \begin{bmatrix} 3 - i & 5 + 2i \\3 - 2i & -3i \end{bmatrix}$

Report an Error

Example Question #32 : The Transpose

$A^{*} = \begin{bmatrix} 3+ 5i & 0 & 0& 0 \\ 2 - 5i & 3+ 6i & 0 & 0 \\1 + 6i &2 - 8 i & 4 + 8 i & 0 \\ 1-2i & 3 - 7i & 2 - 7i & 3- 5i \end{bmatrix}$

True or false: is an upper triangular matrix.

Possible Answers:

False

True

Correct answer:

True

Explanation:

$A^{*}$ is the result of interchanging rows of with columns, then changing each entry to its complex conjugate. Also, is equal to $(A^{*} )^{*} = A$ , so perform the same process on $A^{*}$ :

$A^{*} = \begin{bmatrix} 3+ 5i & 0 & 0& 0 \\ 2 - 5i & 3+ 6i & 0 & 0 \\1 + 6i &2 - 8 i & 4 + 8 i & 0 \\ 1-2i & 3 - 7i & 2 - 7i & 3- 5i \end{bmatrix}$

$(A^{*})^{T} = \begin{bmatrix} 3+ 5i & 2 - 5i & 1 + 6i & 1-2i \\ 0& 3+ 6i & 2 - 8 i & 3 - 7i \\ 0& 0& 4 + 8 i & 2 - 7i \\ 0& 0 & 0 & 3- 5i \end{bmatrix}$

$A= (A^{*})^{*} = \begin{bmatrix} 3- 5i & 2 + 5i & 1 - 6i & 1+2i \\ {\color{Red} \textbf{0} }& 3-6i & 2 + 8 i & 3+ 7i \\ {\color{Red} \textbf{0} } & {\color{Red} \textbf{0} } & 4 - 8 i & 2 +7i \\ {\color{Red} \textbf{0} }& {\color{Red} \textbf{0} } & {\color{Red} \textbf{0} } & 3+ 5i \end{bmatrix}$

A matrix is upper triangular if all elements below its main (upper-left corner to lower right corner) diagonal are equal to 0. These elements in are displayed in red above. Since all of the lower-triangular elements of are zeroes, is upper triangular.

Report an Error

View Linear Algebra Tutors

Mohammad
Certified Tutor

Jamia Millia Islamia, Master of Science, Physics.

View Linear Algebra Tutors

Brienne
Certified Tutor

Georgia Institute of Technology-Main Campus, Bachelor of Science, Industrial Engineering. Spelman College, Bachelor of Scienc...

View Linear Algebra Tutors

Ying
Certified Tutor

Iowa State University, Master of Science, Mathematics.

All Linear Algebra Resources

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

Popular Subjects

Algebra Tutors in Washington DC, ISEE Tutors in New York City, Math Tutors in Dallas Fort Worth, Computer Science Tutors in Chicago, Chemistry Tutors in Seattle, Calculus Tutors in Chicago, Physics Tutors in Atlanta, Chemistry Tutors in San Diego, Algebra Tutors in San Diego, Reading Tutors in Atlanta

Popular Courses & Classes

SSAT Courses & Classes in Phoenix, ACT Courses & Classes in San Diego, SAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Atlanta, SAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Washington DC, GRE Courses & Classes in Miami, GRE Courses & Classes in Seattle, LSAT Courses & Classes in Phoenix, ACT Courses & Classes in Seattle

Popular Test Prep

ACT Test Prep in Seattle, ISEE Test Prep in San Francisco-Bay Area, GRE Test Prep in Seattle, GMAT Test Prep in Chicago, MCAT Test Prep in Denver, GMAT Test Prep in Boston, ACT Test Prep in San Diego, MCAT Test Prep in Boston, GRE Test Prep in Miami, ACT Test Prep in San Francisco-Bay Area

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 #21 : The Transpose

Example Question #51 : Operations And Properties

Example Question #52 : Operations And Properties

Example Question #53 : Operations And Properties

Example Question #51 : Operations And Properties

Example Question #55 : Operations And Properties

Example Question #56 : Operations And Properties

Example Question #56 : Operations And Properties

Example Question #31 : The Transpose

Example Question #32 : The Transpose

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: