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 #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 : Matrices

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

$(5\times 2)$

$(2\times 5)$

$(2\times 2)$

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 #31 : 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:

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

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

$A\times B$ 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

Example Question #41 : Matrix Matrix Product

Find the product $A \times B$ .

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

Possible Answers:

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11\\ 37& 10 \end{bmatrix}$

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

$A \times B$ cannot be multiplied

$A \times B=\begin{bmatrix} -7& -2\\ -8& 15 \\ 5& -1 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11\\ 37& 10 \end{bmatrix}$

Explanation:

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

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

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22}\\ ab_{31}& ab_{32} \end{bmatrix}$

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

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

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22}\\ ab_{31}& ab_{32} \end{bmatrix}$

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

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

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

$ab_{31}=(5)(7)+(-1)(-2)=35+2=37$

$ab_{32}=(5)(-1)+(-1)(5)=-5-5=10$

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11\\ 37& 10 \end{bmatrix}$

Report an Error

Example Question #42 : Matrices

Find the product $A \times B$ .

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

Possible Answers:

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

$A \times B=\begin{bmatrix} 6& 1 &5\\ 2& 8 &1 \end{bmatrix}$

$A \times B$ cannot be multiplied.

$A \times B=\begin{bmatrix} -7& -2 &5\\ -8& 15 &1 \end{bmatrix}$

Correct answer:

$A \times B$ cannot be multiplied.

Explanation:

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

Since is a $2\times3$ matrix and is a $2\times2$ matrix, $n\neq r$ so $(A\times B)$ cannot be multiplied.

Report an Error

Example Question #723 : Linear Algebra

Find the product $A \times B$ .

$A=\begin{bmatrix} -1 &5\\ 4 &1 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -7& -5& 0\\ -8& 5&8 \end{bmatrix}$

$A \times B=\begin{bmatrix} -1& 5& 0\\ 4& 1&8 \end{bmatrix}$

$A \times B$ cannot be multiplied

$A \times B=\begin{bmatrix} -17& 26& 40\\ 26& 1&8 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -17& 26& 40\\ 26& 1&8 \end{bmatrix}$

Explanation:

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

Since is a $2\times2$ matrix and is a $2\times3$ 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

$A=\begin{bmatrix} -1 &5\\ 4 &1 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8 \end{bmatrix}$

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

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

$ab_{11}=(-1)(7)+(5)(-2)=-7-10=-17$

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

$ab_{12}=(-1)(-1)+(5)(5)=1+25=26$

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

$ab_{13}=(-1)(0)+(5)(8)=0+40=40$

$ab_{21}=(4)(7)+(1)(-2)=28-2=26$

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

$ab_{23}=(4)(0)+(1)(8)=0+8=8$

$A \times B=\begin{bmatrix} -17& 26& 40\\ 26& 1&8 \end{bmatrix}$

Report an Error

Example Question #721 : Linear Algebra

Find the product $A \times B$ .

$A=\begin{bmatrix} -1 &5&2\\ 4 &-4&5 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8\\ 1& 2&4 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -7 &-5&0\\ -8 &-20&40 \end{bmatrix}$

$A \times B=\begin{bmatrix} -15& 30& 48\\ 41& -14&-12 \end{bmatrix}$

$A \times B$ cannot be multiplied

$A \times B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -15& 30& 48\\ 41& -14&-12 \end{bmatrix}$

Explanation:

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

Since is a $2\times3$ matrix and is a $3\times3$ 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

$A=\begin{bmatrix} -1 &5&2\\ 4 &-4&5 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8\\ 1& 2&4 \end{bmatrix}$

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

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

$ab_{11}=(-1)(7)+(5)(-2)+(2)(1)=-7-10+2=-15$

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

$ab_{12}=(-1)(-1)+(5)(5)+(2)(2)=1+25+4=30$

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

$ab_{13}=(-1)(0)+(5)(8)+(2)(4)=0+40+8=48$

$ab_{21}=(4)(7)+(-4)(-2)+(5)(1)=28+8+5=41$

$ab_{22}=(4)(-1)+(-4)(5)+(5)(2)=-4-20+10=-14$

$ab_{23}=(4)(0)+(-4)(8)+(5)(4)=0-32+20=-12$

$A \times B=\begin{bmatrix} -15& 30& 48\\ 41& -14&-12 \end{bmatrix}$

Report an Error

View Linear Algebra Tutors

Samuel
Certified Tutor

York University, Bachelor, Physics. York University, Master's/Graduate, Mathematics and Statistics.

View Linear Algebra Tutors

Joseph
Certified Tutor

Lehigh University, Bachelor of Science, Biology, General. Yonsei University, Master of Science, Statistics.

View Linear Algebra Tutors

Mohammad
Certified Tutor

Rutgers University-New Brunswick, Bachelor of Science, Mathematics. Rutgers University-New Brunswick, Master of Science, Stat...

All Linear Algebra Resources

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

Popular Subjects

Spanish Tutors in Denver, French Tutors in Atlanta, Computer Science Tutors in Chicago, SAT Tutors in Washington DC, SAT Tutors in San Diego, MCAT Tutors in Atlanta, ISEE Tutors in Phoenix, Algebra Tutors in Houston, GMAT Tutors in Dallas Fort Worth, GRE Tutors in Dallas Fort Worth

Popular Courses & Classes

SAT Courses & Classes in Miami, MCAT Courses & Classes in Atlanta, SSAT Courses & Classes in Atlanta, GRE Courses & Classes in Phoenix, LSAT Courses & Classes in Boston, Spanish Courses & Classes in Denver, LSAT Courses & Classes in Seattle, LSAT Courses & Classes in Philadelphia, ACT Courses & Classes in Washington DC, SAT Courses & Classes in Seattle

Popular Test Prep

GMAT Test Prep in Phoenix, MCAT Test Prep in Boston, SAT Test Prep in Chicago, LSAT Test Prep in Atlanta, SSAT Test Prep in Dallas Fort Worth, SAT Test Prep in San Diego, ACT Test Prep in New York City, GMAT Test Prep in Houston, ACT Test Prep in Philadelphia, SAT Test Prep in Houston

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 #32 : Matrices

Example Question #32 : Matrices

Example Question #31 : Matrices

Example Question #34 : Matrices

Example Question #35 : Matrix Matrix Product

Example Question #31 : Matrix Matrix Product

Example Question #41 : Matrix Matrix Product

Example Question #42 : Matrices

Example Question #723 : Linear Algebra

Example Question #721 : Linear Algebra

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: