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Q24E

Expert-verifiedFound in: Page 1

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**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.

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.

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.

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.

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.

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