Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : 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 #31 : Matrix Matrix Product

Let $L = \begin{bmatrix} 1& 0\\ 4& 2 \end{bmatrix}$ , $U = \begin{bmatrix} 2&-5 \\ 0& 1 \end{bmatrix}$ , and $A = \begin{bmatrix} 2&-5\\ 8& -18 \end{bmatrix}$ .

True or false: is an example of a valid -factorization of .

Possible Answers:

False, because is not the right kind of matrix.

False, because is not a factorization of .

False, because is not the right kind of matrix.

True

Correct answer:

True

Explanation:

An -factorization is a way of expressing a matrix as a product of two matrices and . For the factorization to be valid:

1) must be a Lower triangular matrix - all elements above its main diagonal (upper left corner to lower right corner) must be "0".

2) must be an Upper triangular matrix - all elements below its main diagonal must be "0".

3) LU = A

The factorization can be seen to satisfy the first two criteria - is lower triangular in that there are no nonzero elements above its main diagonal, and is, analogously, upper triangular. It remains to be shown that LU = A .

Multiply each row in by each column in - add the products of each element in the former by the corresponding element in the latter - as follows:

$LU = \begin{bmatrix} 1& 0\\ 4& 2 \end{bmatrix}\begin{bmatrix} 2&-5 \\ 0& 1 \end{bmatrix}$

$= \begin{bmatrix} 1(2)+ 0(0) & 1(-5)+ 0(1)\\ 4(2)+ 2(0) & 4(-5)+ 2(1) \end{bmatrix}$

$= \begin{bmatrix} 2&-5\\ 8& -18 \end{bmatrix}$

All three criteria are met, and gives a valid -factorization of .

Report an Error

Example Question #31 : Matrix Matrix Product

Let $L = \begin{bmatrix} 1& 0\\ -3& 2 \end{bmatrix}$ , $U = \begin{bmatrix} 2&2 \\ 0& -3 \end{bmatrix}$ , and $A = \begin{bmatrix} 2& 0\\ 0& -6 \end{bmatrix}$

True or false: is an example of a valid -factorization of .

Possible Answers:

True

False, because is not the right kind of matrix.

False, because is not a factorization of .

False, because is not the right kind of matrix.

Correct answer:

False, because is not a factorization of .

Explanation:

An -factorization is a way of expressing a matrix as a product of two matrices and . For the factorization to be valid:

1) must be a Lower triangular matrix - all elements above its main diagonal (upper left corner to lower right corner) must be "0".

2) must be an Upper triangular matrix - all elements below its main diagonal must be "0".

3) LU = A

Multiply each row in by each column in - add the products of each element in the former by the corresponding element in the latter - as follows:

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

$= \begin{bmatrix} 1(2)+ 0(0) & 1(2)+0(-3)\\ (-3)(2)+2(0)& -3(2)+2(-3) \end{bmatrix}$

$= \begin{bmatrix} 2 &2\\-6&-12 \end{bmatrix}$

$LU \ne A$ , so does not give a valid -factorization of .

Report an Error

Example Question #33 : Matrix Matrix Product

True or False: If is a square matrix, and A^2 = 0 , then is either or .

Possible Answers:

True

False

Correct answer:

False

Explanation:

For example, if $A = \begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix}$ , then $A^2 = \begin{bmatrix} 0& 1\\ 0& 0 \end{bmatrix}\begin{bmatrix} 0& 1\\ 0&0 \end{bmatrix} = \begin{bmatrix} 0&0 \\ 0& 0 \end{bmatrix}$ , but itself is not or .

(If represented a single real number, then the question would be true, but since is a matrix, the question is not true anymore.)

Report an Error

Example Question #33 : Matrix Matrix Product

What is dimension criteria to multiply two matrices $(A\times B)$ ?

Possible Answers:

$(A\times B)$ can only be multiplied if the number of rows in equals the number of rows in

$(A\times B)$ can only be multiplied if the number of columns in equals the number of columns in

$(A\times B)$ can only be multiplied if the number of columns in equals the number of rows in

$(A\times B)$ can only be multiplied if the number of rows in equals the number of columns in

Correct answer:

$(A\times B)$ can only be multiplied if the number of columns in equals the number of rows in

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

$(A\times B)$ can only be multiplied if n=r , or the number of columns in equals the number of rows in . Otherwise there is a mismatch, and the two matrices can not be multiplied.

Report an Error

Example Question #32 : Matrices

If is a $3 \times 2$ matrix and is a $2 \times 4$ matrix, can the product $A \times B$ be multiplied? What about $B \times A$ ?

Possible Answers:

$A \times B$ can be multiplied

$B \times A$ cannot be multiplied

$A \times B$ can be multiplied

$B \times A$ can be multiplied

$A \times B$ cannot be multiplied

$B \times A$ cannot be multiplied

$A \times B$ cannot be multiplied

$B \times A$ can be multiplied

Correct answer:

$A \times B$ can be multiplied

$B \times A$ cannot be multiplied

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix, $(A\times B)$ can only be multiplied if n=r .

Since is a $3 \times 2$ matrix and is a $2 \times 4$ matrix, n=r and $A \times B$ can be multiplied.

has rows and has columns, therefore $B \times A$ cannot be multiplied.

Report an Error

Example Question #32 : Matrices

If is a $3 \times 4$ matrix and is a $4 \times 4$ matrix, what are the dimensions of the product $A \times B$ ?

Possible Answers:

$3 \times 3$

$4 \times 4$

$3 \times 4$

$A \times B$ cannot be multiplied

Correct answer:

$3 \times 4$

Explanation:

If is an $(m\times n)$ matrix and is an $(n\times s)$ matrix, the dimensions of $(A\times B)$ are $(m\times s)$ .

In this problem, If is a $3 \times 4$ matrix and is a $4 \times 4$ matrix, so

the dimensions of $(A\times B)$ are $(3\times 4)$ .

Report an Error

Example Question #31 : Matrix Matrix Product

If is a $2 \times 5$ matrix and is a $5 \times 2$ matrix, what are the dimensions of the product $A \times B$ ?

Possible Answers:

$A \times B$ cannot be multiplied

$(2\times 2)$

$(2\times 5)$

$(5\times 2)$

Correct answer:

$(2\times 2)$

Explanation:

If is an $(m\times n)$ matrix and is an $(n\times s)$ matrix, the dimensions of $(A\times B)$ are $(m\times s)$ .

In this problem, If is a $2 \times 5$ matrix and is a $5 \times 2$ matrix, so

the dimensions of $(A\times B)$ are $(2\times 2)$ .

Report an Error

Example Question #34 : Matrices

Find the product $A \times B$ .

$A=\begin{bmatrix} -1 &5\\ 4 &-4 \end{bmatrix}$ , $B=\begin{bmatrix} 2 &-1&5\\ 0 &-7&4 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -2 &-5&5\\ 0 &28&4 \end{bmatrix}$

$A \times B=\begin{bmatrix} -2& -34& 15\\ 8& 24&4 \end{bmatrix}$

$A \times B$ cannot be multiplied

$A \times B=\begin{bmatrix} -1 &5&5\\ 4&-4&4 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -2& -34& 15\\ 8& 24&4 \end{bmatrix}$

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

$(A\times B)$ can only be multiplied if n=r , or the number of columns in equals the number of rows in . Otherwise there is a mismatch, and the two matrices can not be multiplied. $(A\times B)$ will be an $(m\times s)$ matrix

Since is a $2\times2$ matrix and is a $2\times3$ matrix, then $(A\times B)$ can be multiplied and will have the dimensions $2\times3$ .

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}& ab_{13}\\ ab_{21}& ab_{22}& ab_{23} \end{bmatrix}$

To find the product $A \times B$ , you must find the dot product of the rows of and the columns of

$A=\begin{bmatrix} -1 &5\\ 4 &-4 \end{bmatrix}$ , $B=\begin{bmatrix} 2 &-1&5\\ 0 &-7&4 \end{bmatrix}$

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}& ab_{13}\\ ab_{21}& ab_{22}& ab_{23} \end{bmatrix}$

We find $ab_{11}$ by finding the dot product of the row of and column of .

$ab_{11}=(-1)(2)+(5)(0)=-2+0=-2$

We find $ab_{12}$ by finding the dot product of the row of and column of .

$ab_{12}=(-1)(-1)+(5)(-7)=1-35=-34$

We use the same method to find the rest of the matrix values

$ab_{13}=(-1)(5)+(5)(4)=-5+20=15$

$ab_{21}=(4)(2)+(-4)(0)=8+0=8$

$ab_{22}=(4)(-1)+(-4)(-7)=-4+28=24$

$ab_{23}=(4)(5)+(-4)(4)=20-16=4$

$A \times B=\begin{bmatrix} -2& -34& 15\\ 8& 24&4 \end{bmatrix}$

Report an Error

Example Question #35 : Matrix Matrix Product

Find the product $A \times B$ .

$A=\begin{bmatrix} -1& 2\\ 4& 3 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -7& -2\\ -8& 15 \end{bmatrix}$

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11 \end{bmatrix}$

$A\times B$ cannot be multiplied

$A \times B=\begin{bmatrix} 6& 1\\ 2& 8 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11 \end{bmatrix}$

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

Since is a $2\times2$ matrix and is a $2\times2$ matrix, then $(A\times B)$ can be multiplied and will have the dimensions $2\times2$ .

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22} \end{bmatrix}$

To find the product $A \times B$ , you must find the dot product of the rows of and the columns of

$A=\begin{bmatrix} -1& 2\\ 4& 3 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \end{bmatrix}$

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22} \end{bmatrix}$

We find $ab_{11}$ by finding the dot product of the row of and column of .

$ab_{11}=(-1)(7)+(2)(-2)=-7-4=-11$

We find $ab_{12}$ by finding the dot product of the row of and column of .

$ab_{12}=(-1)(-1)+(2)(5)=1+10=11$

We use the same method to find the rest of the matrix values

$ab_{21}=(4)(7)+(3)(-2)=28-6=22$

$ab_{22}=(4)(-1)+(3)(5)=-4+15=11$

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11 \end{bmatrix}$

Report an Error

Example Question #36 : Matrix Matrix Product

Find the product $A \times B$ .

$A=\begin{bmatrix} -1& 2&0\\ 4& 3&6 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \\ 1 & 3 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -1& 2\\ -2& 5 \end{bmatrix}$

$A \times B=\begin{bmatrix} 2& 10\\ 4& 9 \end{bmatrix}$

$A\times B$ cannot be multiplied

$A \times B=\begin{bmatrix} -11& 11\\ 28& 29 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -11& 11\\ 28& 29 \end{bmatrix}$

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

Since is a $2\times3$ matrix and is a $3\times2$ matrix, then $(A\times B)$ can be multiplied and will have the dimensions $2\times2$ .

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22} \end{bmatrix}$

To find the product $A \times B$ , you must find the dot product of the rows of and the columns of

$A=\begin{bmatrix} -1& 2&0\\ 4& 3&6 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \\ 1 & 3 \end{bmatrix}$

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22} \end{bmatrix}$

We find $ab_{11}$ by finding the dot product of the row of and column of .

$ab_{11}=(-1)(7)+(2)(-2)+(0)(1)=-7-4+0=-11$

We find $ab_{12}$ by finding the dot product of the row of and column of .

$ab_{12}=(-1)(-1)+(2)(5)+(0)(3)=1+10+0=11$

We use the same method to find the rest of the matrix values

$ab_{21}=(4)(7)+(3)(-2)+(6)(1)=28-6+6=28$

$ab_{22}=(4)(-1)+(3)(5)+(6)(3)=-4+15+18=29$

$A \times B=\begin{bmatrix} -11& 11\\ 28& 29 \end{bmatrix}$

Report an Error

View Linear Algebra Tutors

Chelsea
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Physics. The University of Texas at Dallas, Master of Science, Bioinf...

View Linear Algebra Tutors

Nima
Certified Tutor

Sahand University of Technology, Master of Science, Electrical Engineering. Institut national de la recherche scientifique, D...

View Linear Algebra Tutors

Liban
Certified Tutor

Emory University, Bachelor of Science, Mathematics/Economics.

All Linear Algebra Resources

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

Popular Subjects

Algebra Tutors in Phoenix, SSAT Tutors in Philadelphia, English Tutors in Washington DC, Biology Tutors in Phoenix, GRE Tutors in Chicago, ACT Tutors in Washington DC, Calculus Tutors in New York City, GMAT Tutors in Boston, Statistics Tutors in Los Angeles, French Tutors in Atlanta

Popular Courses & Classes

SSAT Courses & Classes in Seattle, GMAT Courses & Classes in Boston, GMAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in San Diego, ISEE Courses & Classes in Atlanta, LSAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in New York City, SSAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in New York City, SSAT Courses & Classes in Los Angeles

Popular Test Prep

GMAT Test Prep in San Diego, LSAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Los Angeles, SSAT Test Prep in Boston, ISEE Test Prep in San Francisco-Bay Area, GMAT Test Prep in Boston, GRE Test Prep in Boston, MCAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Atlanta, ISEE 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 : Matrices

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #31 : Matrix Matrix Product

Example Question #31 : Matrix Matrix Product

Example Question #33 : Matrix Matrix Product

Example Question #33 : Matrix Matrix Product

Example Question #32 : Matrices

Example Question #32 : Matrices

Example Question #31 : Matrix Matrix Product

Example Question #34 : Matrices

Example Question #35 : Matrix Matrix Product

Example Question #36 : Matrix Matrix Product

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: