Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Matrices

Study concepts, example questions & explanations for Calculus 3

Create An Account

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2701 : Calculus 3

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix}-8 &1 &0 \\-8 &-5 &-9 \\2 &-7 &7 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\-5 \\-8 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -39\\-115 \\82 \end{vmatrix}$

$\begin{vmatrix} 86\\25 \\44 \end{vmatrix}$

$\begin{vmatrix} 11\\-5 \\4 \end{vmatrix}$

$\begin{vmatrix} -23\\18 \\-42 \end{vmatrix}$

$\begin{vmatrix} -37\\65 \\-13 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -37\\65 \\-13 \end{vmatrix}$

Explanation:

In order to multiply two matrices, $A\times b$ , the respective dimensions of each must be of the form $m \times n$ and $n \times p$ to create an $m\times p$ (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

$(A\times b \neq b \times A)$

For a multiplication of the form

$\begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}$

The resulting matrix is

$\begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}$

The notation may be daunting but numerical examples may elucidate.

We're told that

$A=\begin{vmatrix}-8 &1 &0 \\-8 &-5 &-9 \\2 &-7 &7 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\-5 \\-8 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -32-5+0\\-32+25+72 \\8+35-56 \end{vmatrix}=\begin{vmatrix} -37\\65 \\-13 \end{vmatrix}$

Report an Error

Example Question #2702 : Calculus 3

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix}9 &7 &9 \\7 &-5 &-6 \\-1 &3 &-9 \end{vmatrix}$ and $b=\begin{vmatrix} -7\\-7 \\8 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix}40 \\62 \\-86 \end{vmatrix}$

$\begin{vmatrix}-40 \\-62 \\86 \end{vmatrix}$

$\begin{vmatrix}-40 \\62 \\-86 \end{vmatrix}$

$\begin{vmatrix}-40 \\-62 \\-86 \end{vmatrix}$

$\begin{vmatrix}40 \\62 \\86 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}-40 \\-62 \\-86 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that

$A=\begin{vmatrix}9 &7 &9 \\7 &-5 &-6 \\-1 &3 &-9 \end{vmatrix}$ and $b=\begin{vmatrix} -7\\-7 \\8 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -63-49+72\\-49+35-48 \\7-21-72 \end{vmatrix}=\begin{vmatrix}-40 \\-62 \\-86 \end{vmatrix}$

Report an Error

Example Question #2701 : Calculus 3

Calculate the determinant of .

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

Possible Answers:

$\det(A)=-2$

$\det(A)=1$

$\det(A)=2$

$\det(A)=-1$

$\det(A)=0$

Correct answer:

$\det(A)=-2$

Explanation:

In order to find the determinant, we need to multiply the main diagonal components and then subtract the off main diagonal components.

$\det(A)=1(4)-(3)(2)=4-6=-2$

Report an Error

Example Question #704 : Vectors And Vector Operations

Find the product of the two matrices:

$\mathbf{A}\mathbf{B}$

Where

$\mathbf{A}=\begin{pmatrix} 3&1 \\ 2&-2 \end{pmatrix}$

and

$\mathbf{B}=\begin{pmatrix} -1&4 \\ 0&5 \end{pmatrix}$

Possible Answers:

$\begin{pmatrix} 5&10 \\ -9&-10 \end{pmatrix}$

$\begin{pmatrix} 3&-2 \\ 17&-2 \end{pmatrix}$

$\begin{pmatrix} -3&17 \\ -2&-2 \end{pmatrix}$

$\begin{pmatrix} 5&-9 \\ 10&-10 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} -3&17 \\ -2&-2 \end{pmatrix}$

Explanation:

$\mathbf{AB}=\begin{pmatrix} 3&1 \\ 2&-2 \end{pmatrix}\textup{}$ $\begin{pmatrix} -1&4 \\ 0&5 \end{pmatrix}$

$=\begin{pmatrix} (3)(-1)+(1)(0)&(3)(4)+(1)(5) \\ (2)(-1)+(-2)(0)&(2)(4)+(-2)(5) \end{pmatrix}$

$=\begin{pmatrix} (-3+0)&(12+5) \\ (-2+0)&(8-10) \end{pmatrix}=\begin{pmatrix} -3&17 \\ -2&-2 \end{pmatrix}$

Report an Error

Example Question #115 : Matrices

Evaluate the following matrix operation:

$2\mathbf{A}+4\mathbf{B}$

where

$\mathbf{A}=\begin{pmatrix} 1&4 \\ -3&0 \end{pmatrix}, \, \;\mathbf{B}=\begin{pmatrix} 2&-2 \\ 1&-1 \end{pmatrix}$

Possible Answers:

$\begin{pmatrix} 8&12 \\ -10&-2 \end{pmatrix}$

$\begin{pmatrix} 10&0 \\ -2&-4 \end{pmatrix}$

$\begin{pmatrix} -6&16 \\ -10&4 \end{pmatrix}$

$\begin{pmatrix} -1&6 \\ -4&1 \end{pmatrix}$

$\begin{pmatrix} 2&-8 \\ -2&-1 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} 10&0 \\ -2&-4 \end{pmatrix}$

Explanation:

$2\mathbf{A}+4\mathbf{B}=2\begin{pmatrix} 1&4 \\ -3&0 \end{pmatrix}+4 \begin{pmatrix} 2&-2 \\ 1&-1 \end{pmatrix}= \begin{pmatrix} 2&8 \\ -6&0 \end{pmatrix}+ \begin{pmatrix} 8&-8 \\ 4&-4 \end{pmatrix}$

$=\begin{pmatrix} 2+8&8-8 \\ -6+4&0-4 \end{pmatrix}= \begin{pmatrix} 10&0 \\ -2&-4 \end{pmatrix}$

Report an Error

Example Question #111 : Matrices

Find the determinant of the matrix A:

$\mathbf{A}= \begin{pmatrix} 7&4 \\ 5&2 \end{pmatrix}$

Possible Answers:

-6

-24

Correct answer:

-6

Explanation:

The determinant of a matrix

$\begin{pmatrix} a&b \\ c&d \end{pmatrix}$

is defined as:

ad-bc

Here, that becomes:

(7)(2)-4(5)=14-20=-6

Report an Error

Example Question #113 : Matrices

$A=\begin{pmatrix} 1&4 \\ 5&7 \end{pmatrix}$

What is A^t ?

Possible Answers:

$A^t=\begin{pmatrix} 4&7 \\ 1& 5 \end{pmatrix}$

$A^t=\begin{pmatrix} 1&5 \\ 4& 7 \end{pmatrix}$

$A^t=\begin{pmatrix} 1&-4 \\ -5& 7 \end{pmatrix}$

$A^t=\begin{pmatrix} 5&1 \\ 7& 4 \end{pmatrix}$

Correct answer:

$A^t=\begin{pmatrix} 1&5 \\ 4& 7 \end{pmatrix}$

Explanation:

To find A^t , we write the rows of as columns.

Hence,

$A^t=\begin{pmatrix} 1&5 \\ 4& 7 \end{pmatrix}$

Report an Error

Example Question #112 : Matrices

$A=\begin{pmatrix} 4&3 \\ 2&5 \end{pmatrix}$

What is A^t ?

Possible Answers:

$A^t=\begin{pmatrix} 4&2 \\ 3& 5 \end{pmatrix}$

$A^t=\begin{pmatrix} 3&5 \\ 4&2 \end{pmatrix}$

$A^t=\begin{pmatrix} 4&-3 \\ -2&5 \end{pmatrix}$

$A^t=\begin{pmatrix} 2&4 \\ 5&3 \end{pmatrix}$

Correct answer:

$A^t=\begin{pmatrix} 4&2 \\ 3& 5 \end{pmatrix}$

Explanation:

To find A^t , we write the rows of as columns.

Hence,

$A^t=\begin{pmatrix} 4&2 \\ 3& 5 \end{pmatrix}$

Report an Error

Example Question #113 : Matrices

Find the determinant of the matrix.

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

Possible Answers:

Correct answer:

Explanation:

The formula for the determinant of a 3x3 matrix

$A=\begin{bmatrix} a&b &c \\ d &e & f\\ g&h &i \end{bmatrix}$ is

$\left | A\right |=a(ei-hf)-b(di-gf)+c(dh-ge)$ .

Using the matrix we were given, we get

1(3-4)-4(2-0)+5(2-0)=1 .

Report an Error

Example Question #114 : Matrices

Perform the matrix operation.

$\begin{bmatrix} 2& 3\\6 & 7 \end{bmatrix}+\begin{bmatrix} 1& 5\\ 0& 1 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 3& 8\\ 6&8 \end{bmatrix}$

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

$\begin{bmatrix} 9&1 \\ 2&3 \end{bmatrix}$

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

Correct answer:

$\begin{bmatrix} 3& 8\\ 6&8 \end{bmatrix}$

Explanation:

The formula for adding a pair of 2x2 matrices is

$\begin{bmatrix} a& b\\ c&d \end{bmatrix}+\begin{bmatrix} e&f \\ g &h \end{bmatrix}=\begin{bmatrix} (a+e)&(b+f) \\ (c+g)& (d+h) \end{bmatrix}$ .

Using the matrices in the problem statement, we get

$\begin{bmatrix} (2+1)&(3+5) \\ (6+0)&(7+1) \end{bmatrix}=\begin{bmatrix} 3&8 \\ 6& 8 \end{bmatrix}$

Report an Error

View Calculus 3 Tutors

Eric
Certified Tutor

The University of Texas System Office, Bachelor of Science, Mathematics and Statistics.

View Calculus 3 Tutors

Aaron
Certified Tutor

Tulsa Community College, Associate in Science, Mechanical Engineering.

View Calculus 3 Tutors

Joshua
Certified Tutor

The Ohio State University, Bachelor of Science, Mathematics.

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SSAT Tutors in Philadelphia, French Tutors in Phoenix, GRE Tutors in Chicago, ACT Tutors in Boston, Spanish Tutors in Miami, ISEE Tutors in New York City, Math Tutors in Boston, English Tutors in San Diego, Math Tutors in Denver, ACT Tutors in Denver

Popular Courses & Classes

LSAT Courses & Classes in Chicago, MCAT Courses & Classes in Atlanta, Spanish Courses & Classes in Miami, ACT Courses & Classes in Seattle, GRE Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Los Angeles, Spanish Courses & Classes in Phoenix, MCAT Courses & Classes in Philadelphia, GRE Courses & Classes in Houston

Popular Test Prep

LSAT Test Prep in Washington DC, ISEE Test Prep in New York City, MCAT Test Prep in Washington DC, LSAT Test Prep in Atlanta, GRE Test Prep in Philadelphia, MCAT Test Prep in Miami, ISEE Test Prep in Atlanta, GMAT Test Prep in San Diego, GRE Test Prep in Denver, GMAT 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

Calculus 3 : Matrices

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #2701 : Calculus 3

Example Question #2702 : Calculus 3

Example Question #2701 : Calculus 3

Example Question #704 : Vectors And Vector Operations

Example Question #115 : Matrices

Example Question #111 : Matrices

Example Question #113 : Matrices

Example Question #112 : Matrices

Example Question #113 : Matrices

Example Question #114 : Matrices

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: