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Q24E

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Linear Algebra and its Applications
Found in: Page 1
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer.(If true, give the approximate location where a similar statement appears, or refer to a definition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.

24.

a. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.

b. Two matrices are row equivalent if they have the same number of rows.

c. An inconsistent system has more than one solution.

d. Two linear systems are equivalent if they have the same solution set.

Statements (a) and (d) are true, while (b) and (c) are false.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Explanation for part (a)

Performing elementary row operations to an augmented matrix is similar to algebraically manipulating the corresponding system to obtain a linear system with the same solution (refer to page 7).

Example: Consider the augmented matrix of a linear system, \(\left[ {\begin{array}{*{20}{c}}{ - 12}&9&7\\9&{ - 12}&6\end{array}} \right]\). A row operation applied on this matrix can be reversed by applying another operation. It means the elementary row operations on the augmented matrix do not change the solution set of the associated linear system.

Hence, the statement is true.

Step 2: Explanation for part (b)

If two matrices are row equivalents, a sequence of elementary row operations can be performed on one to obtain the other (refer to page 6).

For example:

Consider two matrices \(A = \left[ {\begin{array}{*{20}{c}}{ - 2}&9\\4&{ - 11}\end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}}{11}&{ - 5}\\7&8\end{array}} \right]\). These matrices have the same number of rows but are not row equivalent because elementary operations on one matrix do not give the other matrix.

Hence, the statement is false.

Step 3: Explanation for part (c)

A system with no solution (i.e., lines are parallel and do not intersect each other) is said to be inconsistent (refer to page 4).

For example:

Consider the system of linear equations \(\left[ {\begin{array}{*{20}{c}}1&{ - 1}&8\\5&{ - 5}&{25}\end{array}} \right]\). The determinant of the augmented matrix becomes zero while calculating, which means the system has no solution.

Hence, the statement is false.

Step 4: Explanation for part (d)

Equivalent systems are the systems of equations that have the same solution (refer to page 7).

For example:

Let two linear systems of equations be \(\left[ {\begin{array}{*{20}{c}}{ - 12}&9&7\\9&{ - 12}&6\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 12}&9&7\\3&{ - 4}&2\end{array}} \right]\). An elementary row operation (i.e., row two becomes thrice) is applied on row two of the second system. Thus, the second system becomes \(\left[ {\begin{array}{*{20}{c}}{ - 12}&9&7\\9&{ - 12}&6\end{array}} \right]\).

This system is the same as the first one. These two systems are said to be equivalent, with the same solution set.

Hence, the statement is true.

Most popular questions for Math Textbooks

Let \(u = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\) and \(v = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\). Show that \(\left[ {\begin{array}{*{20}{c}}h\\k\end{array}} \right]\) is in Span \(\left\{ {u,v} \right\}\) for all \(h\) and\(k\).

In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).

12.

Find the general solutions of the systems whose augmented matrices are given in Exercises 10.

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

Question: Determine whether the statements that follow are true or false, and justify your answer.

16: There exists a 2x2 matrix such that

A[11]=[12] and A[22]=[21] .

Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.

18. \(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&{ - 2}\\5&h&{ - 7}\end{array}} \right]\)
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