/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q21E Mark each statement True or Fals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q21E (page 1)

Mark each statement True or False. Justify each answer.

a. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.

b. The row reduction algorithm applies only to augmented matrices for a linear system.

c. A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.

d. Finding a parametric description of the solution set of a linear system is the same as solving the system.

e. If one row in an echelon form of an augmented matrix is \(\left( {\begin{array}{*{20}{c}}0&0&0&5&0\end{array}} \right)\), then the associated linear system is inconsistent.

Short Answer

Expert verified

a. False

b. False

c. True

d. True

e. False

Step by step solution

01

Analyzing statement (a)

There can only be one reduced echelon form in a matrix. According to the uniqueness of the reduced echelon form theorem, the reduced echelon form of a matrix is unique.

So, the given statement is false.

02

Analyzing statement (b)

The row reduction algorithm can be used to reduce any matrix.

Hence, the given statement is false.

03

Analyzing statement (c)

Basic variables correspond to columns with 1 as the leading entry (pivot columns).

Hence, the given statement is true.

04

Analyzing statement (d)

The solution of a system of equations with infinite solutions can be written in the parametric form by assuming the free variables as parameters, as explained in the item "parametric description of solution sets."

Hence, the given statement is true.

05

Analyzing statement (e)

There are five columns and four variables. The variable of the provided column has a non-zero item, while the constant (last) column does not. As a result, the equation only signifies that the fourth-position variable is zero, not that the corresponding linear system is incoherent.

Hence, the given statement is false.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 5-8, write a matrix equation that determines the loop currents. [M] If MATLAB or another matrix program is available, solve the system for the loop currents.

A large apartment building is to be built using modular construction techiniques. The arrangement of apartment on any particular floor is to be chosen from one of three basic floor plans. Plan A has 18 apartments on one floor, including 3 three bedroom units, and 8 one bedroom units. 7 two bedroom units and 8 one bedroom units. Each floor of plan B includes 4 three bedroom units, 4 two bedroom units, and 8 one bedroom units. Each floor of plan C includes 5 three bedroom units, 3 two bedroom units, and 9 one bedroom units. Suppose the building contains a total of \({x_{\bf{1}}}\) floors of plan A, \({x_2}\) floor of plans B, and \({x_{\bf{3}}}\) floors of plan C.

a. What interpretation can be given to the vector \({x_{\bf{1}}}\left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{7}}\\{\bf{8}}\end{aligned}} \right)\)?

b. Write a formal linear combination of vectors that expresses the total numbers of three-, two-, and one-bedroom apartments contained in the building.

c. (M) Is it possible to design the building with exactly 66 three bedroom units, 74 two bedrooms units, and 136 one bedroom units? If so, is there more than one way to do it? Explain your answer.

Give a geometric description of Span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors in Exercise 16.

Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.

5. \(\left( {\begin{aligned}{*{20}{c}}1&{ - 4}&5&0&7\\0&1&{ - 3}&0&6\\0&0&1&0&2\\0&0&0&1&{ - 5}\end{aligned}} \right)\)

Let \(A = \left[ {\begin{array}{*{20}{c}}2&0&6\\{ - 1}&8&5\\1&{ - 2}&1\end{array}} \right]\), let \(b = \left[ {\begin{array}{*{20}{c}}{10}\\3\\3\end{array}} \right]\) , and let \(W\) be the set of all linear combinations of the columns of \(A\).

  1. Is \(b\) in \(W\)?
  2. Show that the third column of \(A\) is in \(W\).
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.