Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : Linear Equations

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

← Previous 1 2 3 4 5 6 7 8 9 … 12 13 Next →

Example Question #21 : Reduced Row Echelon Form And Row Operations

True or false:

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

is an elementary matrix.

Possible Answers:

False

True

Correct answer:

False

Explanation:

An elementary matrix is one that can be formed by performing a single row operation on the (here, four-by-four) identity matrix

$I = \begin{bmatrix}1 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\0 &0&0&1\end{bmatrix}$

A matrix qualifies as an elementary matrix if and only if it differs from in exactly one of three ways:

1) Exactly two rows have changed position with each other.

2) Exactly one of its diagonal elements - the element in Row 1 and Column 1, Row 2 and Column 2, and so forth - is equal to a nonzero number other than 1.

3) Exactly one of its nondiagonal elements is equal to a nonzero number.

$\begin{bmatrix} \frac{1}{2}&0 &0&0 \\0 & \frac{1}{3}&0&0 \\ 0 &0 & \frac{1}{4} &0 \\0 &0&0& \frac{1}{5}\end{bmatrix}$ violates these criteria because all four of its diagonal elements are equal to nonzero numbers other than 1.

Report an Error

Example Question #21 : Reduced Row Echelon Form And Row Operations

True or false:

$\begin{bmatrix} 0.000001 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\0 &0&0&1\end{bmatrix}$

is an elementary matrix.

Possible Answers:

False

True

Correct answer:

True

Explanation:

An elementary matrix is one that can be formed by performing a single row operation on the (here, four-by-four) identity matrix

$I = \begin{bmatrix}1 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\0 &0&0&1\end{bmatrix}$

A matrix qualifies as an elementary matrix if and only if it differs from in exactly one of three ways:

1) Exactly two rows have changed position with each other.

2) Exactly one of its diagonal elements - the element in Row 1 and Column 1, Row 2 and Column 2, and so forth - is equal to a nonzero number other than 1.

3) Exactly one of its nondiagonal elements is equal to a nonzero number.

$\begin{bmatrix} 0.000001 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\0 &0&0&1\end{bmatrix}$ satisfies these criteria in that it differs from only in that one diagonal element (Row 1, Column 1) has been changed to the nonzero number 0.000001.

Report an Error

Example Question #21 : Reduced Row Echelon Form And Row Operations

True or false:

$\begin{bmatrix}1 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\1 & 1& 1&1\end{bmatrix}$

is an elementary matrix.

Possible Answers:

True

False

Correct answer:

False

Explanation:

An elementary matrix is one that can be formed by performing a single row operation on the (here, four-by-four) identity matrix

$I = \begin{bmatrix}1 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\0 &0&0&1\end{bmatrix}$

A matrix qualifies as an elementary matrix if and only if it differs from in exactly one of three ways:

1) Exactly two rows have changed position with each other.

2) Exactly one of its diagonal elements - the element in Row 1 and Column 1, Row 2 and Column 2, and so forth - is equal to a nonzero number other than 1.

3) Exactly one of its nondiagonal elements is equal to a nonzero number.

$\begin{bmatrix}1 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\1 & 1& 1&1\end{bmatrix}$ violates these criteria because three of its nondiagonal elements - the first three elements in its fourth row - are equal to nonzero numbers.

Report an Error

Example Question #24 : Reduced Row Echelon Form And Row Operations

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

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

Which of the following matrices is equal to ?

Possible Answers:

$\begin{bmatrix} -2&-2 &4 & 5& -2 &1\\ -4&3 & 0& 2& 1 & 2\\ -4& -1& 2& 1 &2&0 \\6 & -4& 1& 2& -3& 6\\2 &-2& 3& 1& 0& 3 \end{bmatrix}$

$\begin{bmatrix} 1&-2 &4 & 5& -2 &1\\ 2&3 & 0& 2& 1 & 2\\ 2& -1& 2& 1 &2&0 \\-5 & 0& -7& -8& 1& 4\\ -1 &-2& 3& 1& 0& 3 \end{bmatrix}$

$\begin{bmatrix} 1&-2 &4 & 5& -2 &1\\ 2&3 & 0& 2& 1 & 2\\ 2& -1& 2& 1 &2&0 \\6 &8& -2& -4& 6& -12 \\ -1 &-2& 3& 1& 0& 3 \end{bmatrix}$

$\begin{bmatrix} 1&-2 &4 & 5& -2 &1\\ 2&3 & 0& 2& 1 & 2\\ 2& -1& 2& 1 &2&0 \\6 &8& -2& -4& 6& -12\\ -1 &-2& 3& 1& 0& 3 \end{bmatrix}$

$\begin{bmatrix} -9&-2 &4 & 5& -2 &1\\ -2&3 & 0& 2& 1 & 2\\ 0& -1& 2& 1 &2&0 \\-7 & -4& 1& 2& -3& 6\\ -3 &-2& 3& 1& 0& 3 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1&-2 &4 & 5& -2 &1\\ 2&3 & 0& 2& 1 & 2\\ 2& -1& 2& 1 &2&0 \\-5 & 0& -7& -8& 1& 4\\ -1 &-2& 3& 1& 0& 3 \end{bmatrix}$

Explanation:

is an elementary matrix in that it can be formed from the (five-by-five) identity matrix by one row operation. differs from in that appears instead of 0 in Row 4, Column 1; therefore, the row operation that has been performed on to obtain is

$-2R1+ R4 \rightarrow R4$

Premultiplying by has the effect of performing the same row operation on , so, to each element in row 4 of , add times the corresponding element of , while leaving all other elements intact:

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

$AB=\begin{bmatrix} 1&-2 &4 & 5& -2 &1\\ 2&3 & 0& 2& 1 & 2\\ 2& -1& 2& 1 &2&0 \\-3+ (-2)(1) & -4+ (-2) (-2)& 1+ (-2)4& 2+(-2)5& -3+ (-2)(-2)& 6+(-2)1\\ -1 &-2& 3& 1& 0& 3 \end{bmatrix}$

$AB=\begin{bmatrix} 1&-2 &4 & 5& -2 &1\\ 2&3 & 0& 2& 1 & 2\\ 2& -1& 2& 1 &2&0 \\-5 & 0& -7& -8& 1& 4\\ -1 &-2& 3& 1& 0& 3 \end{bmatrix}$

Report an Error

Example Question #25 : Reduced Row Echelon Form And Row Operations

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

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

Which of the following matrices is equal to ?

Possible Answers:

$\begin{bmatrix}0 &-2& 3& 1& -1& 3 \\ 1&3 & 0& 2& 2 & 2\\ 2& -1& 2& 1 &2&0 \\-3 & -4& 1& 2& -3& 6\\ -2&-2 &4 & 5& 1 &1\end{bmatrix}$

$\begin{bmatrix}-1 &-2& 3& 1& 0& 3 \\ 2&3 & 0& 2& 1 & 2\\ 2& -1& 2& 1 &2&0 \\-3 & -4& 1& 2& -3& 6\\ 1&-2 &4 & 5& -2 &1 \end{bmatrix}$

$\begin{bmatrix} -2&-2 &4 & 5& 1 &1\\ 1&3 & 0& 2& 2 & 2\\ 2& -1& 2& 1 &2&0 \\-3 & -4& 1& 2& -3& 6\\0 &-2& 3& 1& -1& 3 \end{bmatrix}$

$\begin{bmatrix} 1&-2 &4 & 5& -2 &1\\ 2&3 & 0& 2& 1 & 2\\ 0& -1& 2& 1 &2&2 \\6& -4& 1& 2& -3& -3\\ 3 &-2& 3& 1& 0& -1 \end{bmatrix}$

Correct answer:

$\begin{bmatrix}-1 &-2& 3& 1& 0& 3 \\ 2&3 & 0& 2& 1 & 2\\ 2& -1& 2& 1 &2&0 \\-3 & -4& 1& 2& -3& 6\\ 1&-2 &4 & 5& -2 &1 \end{bmatrix}$

Explanation:

is an elementary matrix in that it can be formed from the (five-by-five) identity matrix by one row operation. differs from in that Row 1 and Row 5 have been switched; therefore, the row operation that has been performed on to obtain is $R1 \leftrightarrow R5$ .

Premultiplying by has the effect of performing the same row operation on , so, in , switch Row 1 and Row 5, while leaving all other elements intact:

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

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

the correct choice.

Report an Error

Example Question #21 : Linear Equations

True or false:

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

is an example of a matrix in reduced row-echelon form.

Possible Answers:

False

True

Correct answer:

False

Explanation:

A matrix is considered to be in reduced row-echelon form if and only if it meets all of the following criteria:

1) Any rows comprising only "0" elements must be gathered at the bottom of the matrix.

2) The first nonzero element in each other row must be a "1".

3) The leading "1" in each row must be to the right of the leading "1" of the above row.

The matrix

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

satisfies the first two criteria. However, the third condition is violated, as the leading "1" in the third row is directly under that in the row above it. Therefore, this matrix is not in reduced row-echelon form.

Report an Error

Example Question #22 : Linear Equations

True or false:

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

is an example of a matrix in reduced row-echelon form.

Possible Answers:

True

False

Correct answer:

True

Explanation:

A matrix is considered to be in reduced row-echelon form if and only if it meets all of the following criteria:

1) Any rows comprising only "0" elements must be gathered at the bottom of the matrix.

2) The first nonzero element in each other row must be a "1".

3) The leading "1" in each row must be to the right of the leading "1" of the above row.

The matrix

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

can be seen to satisfy all three conditions and it is therefore in reduced row-echelon form.

Report an Error

Example Question #21 : Linear Equations

True or false:

$\begin{bmatrix} 1& 100 & 200 & 300 \\ 0& 0 & 1 & - 500 \end{bmatrix}$

is an example of a matrix in reduced row-echelon form.

Possible Answers:

True

False

Correct answer:

True

Explanation:

A matrix is considered to be in reduced row-echelon form if and only if it meets all of the following criteria:

1) Any rows comprising only "0" elements must be gathered at the bottom of the matrix.

2) The first nonzero element in each other row must be a "1".

3) The leading "1" in each row must be to the right of the leading "1" of the above row.

The matrix

$\begin{bmatrix} 1& 100 & 200 & 300 \\ 0& 0 & 1 & - 500 \end{bmatrix}$

can be seen to satisfy all three conditions and it is therefore in reduced row-echelon form.

Report an Error

Example Question #29 : Reduced Row Echelon Form And Row Operations

True or false:

$\begin{bmatrix}1 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\1,000,000 &0&0&1\end{bmatrix}$

is an elementary matrix.

Possible Answers:

False

True

Correct answer:

True

Explanation:

An elementary matrix is one that can be formed by performing a single row operation on the (here, four-by-four) identity matrix

$I = \begin{bmatrix}1 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\0 &0&0&1\end{bmatrix}$

A matrix qualifies as an elementary matrix if and only if it differs from in exactly one of three ways:

1) Exactly two rows have changed position with each other.

2) Exactly one of its elements on its main diagonal - upper left to lower right - is equal to a nonzero number other than 1.

3) Exactly one of its nondiagonal elements is equal to a nonzero number.

$\begin{bmatrix}1 &0 &0&0 \\0 &1 &0&0 \\ 0 &0 &1 &0 \\1,000,000 &0&0&1\end{bmatrix}$ satisfies this condition in that it differs from in only one of these ways - one non-diagonal element (Row 4, Column 1) has been changed to a nonzero number. This is an elementary matrix.

Report an Error

Example Question #24 : Linear Equations

True or false:

The matrix

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

is an example of a matrix in reduced row-echelon form.

Possible Answers:

False

True

Correct answer:

False

Explanation:

A matrix is considered to be in reduced row-echelon form if and only if it meets all of the following criteria:

1) Any rows comprising only "0" elements must be gathered at the bottom of the matrix.

2) The first nonzero element in each other row must be a "1".

3) The leading "1" in each row must be to the right of the leading "1" of the above row.

The matrix

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

violates the first condition, as the only zero row in this matrix is above a nonzero row. This matrix is not in reduced row-echelon form.

Report an Error

← Previous 1 2 3 4 5 6 7 8 9 … 12 13 Next →

View Linear Algebra Tutors

Maziar
Certified Tutor

Tehran Polytechnic, Mechanical Engineer, Mechanical Engineering. New Mexico State University-Main Campus, Doctor of Philosoph...

View Linear Algebra Tutors

Liban
Certified Tutor

Emory University, Bachelor of Science, Mathematics/Economics.

View Linear Algebra Tutors

Christopher
Certified Tutor

Arizona State University, Bachelor of Science, Mathematics.

All Linear Algebra Resources

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

Popular Subjects

Computer Science Tutors in Atlanta, ISEE Tutors in Los Angeles, GRE Tutors in Denver, GMAT Tutors in Seattle, SAT Tutors in Phoenix, Spanish Tutors in Chicago, SAT Tutors in Houston, Statistics Tutors in Atlanta, MCAT Tutors in Philadelphia, GRE Tutors in Washington DC

Popular Courses & Classes

ACT Courses & Classes in San Diego, ISEE Courses & Classes in Phoenix, Spanish Courses & Classes in New York City, MCAT Courses & Classes in Washington DC, SSAT Courses & Classes in New York City, LSAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Miami, SAT Courses & Classes in Washington DC, SSAT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Los Angeles

Popular Test Prep

LSAT Test Prep in Houston, LSAT Test Prep in Los Angeles, GRE Test Prep in Washington DC, MCAT Test Prep in Washington DC, ACT Test Prep in Denver, GRE Test Prep in Houston, GRE Test Prep in New York City, ISEE Test Prep in San Francisco-Bay Area, MCAT Test Prep in Miami, LSAT Test Prep in Dallas Fort Worth

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 Equations

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #21 : Reduced Row Echelon Form And Row Operations

Example Question #21 : Reduced Row Echelon Form And Row Operations

Example Question #21 : Reduced Row Echelon Form And Row Operations

Example Question #24 : Reduced Row Echelon Form And Row Operations

Example Question #25 : Reduced Row Echelon Form And Row Operations

Example Question #21 : Linear Equations

Example Question #22 : Linear Equations

Example Question #21 : Linear Equations

Example Question #29 : Reduced Row Echelon Form And Row Operations

Example Question #24 : Linear Equations

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: