/*! 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} Q7E In Exercises 7鈥�10, the augment... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q7E (page 1)

In Exercises 7鈥�10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.

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

Short Answer

Expert verified

The linear system has no solution.

Step by step solution

01

Rewrite the augmented matrix

The augmented matrix of a linear system is given as

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

02

Perform elementary row operations

A basic principle states that row operations do not affect the solution set of a linear system.

Ordinarily, the next step would be to interchange the third row and the fourth row to have 1 in the third row and the third column.

However, in this case, all three elements of the third row of the augmented matrix are zero.

03

Convert the third row into the equation form

The third row can be written in the equation form as

\(\begin{aligned}{c}0{x_1} + 0{x_2} + 0{x_3} = 1\\ \Rightarrow 0 = 1.\end{aligned}\)

This is a contradiction and is not a possible condition.

04

Conclusion

All elements of the third row of the matrix are zero, and it is well known that zero never equals one.

Thus, the solution set is empty, or the given system of linear equations has no solution.

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 7-12, describe all solutions of \(Ax = 0\) in parametric vector form, where \(A\) is row equivalent to the given matrix.

10. \(\left[ {\begin{array}{*{20}{c}}1&3&0&{ - 4}\\2&6&0&{ - 8}\end{array}} \right]\)

Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

30.\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&{ - 2}&6\\0&{ - 5}&9\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&1&{ - 3}\\0&{ - 5}&9\end{array}} \right]\)

Construct three different augmented matrices for linear systems whose solution set is \({x_1} = - 2,{x_2} = 1,{x_3} = 0\).

Let \({\bf{u}}\) and \({\bf{v}}\) be vectors in\({\mathbb{R}^{\bf{n}}}\). It can be shown that the set \({\bf{P}}\) of all points in the parallelogram determined by \({\bf{u}}\) and \({\bf{v}}\) has the form \({\bf{av}} + {\bf{bv}}\), for \({\bf{0}} \le {\bf{a}} \le {\bf{1}}\), \({\bf{0}} \le {\bf{b}} \le {\bf{1}}\). Let \({\bf{T}}:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{n}}}\) be a linear transformation. Explain why the image of a point in \({\bf{P}}\) under the transformation \({\bf{T}}\) lies in the parallelogram determined by \({\bf{T}}\left( {\bf{u}} \right)\) and \({\bf{T}}\left( {\bf{v}} \right)\).

In Exercises 5, write a system of equations that is equivalent to the given vector equation.

5. \({x_1}\left[ {\begin{array}{*{20}{c}}6\\{ - 1}\\5\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{ - 3}\\4\\0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.