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Q19E

Expert-verified

Linear Algebra and its Applications

Found in: Page 165

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 Exercise 19-24, explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant.

\(\left[ {\begin{aligned}{{20}{c}}a&b\\c&d\end{aligned}} \right],\left[ {\begin{aligned}{{20}{c}}c&d\\a&b\end{aligned}} \right]\)

The row operation swaps row 1 and 2 of the matrix and reverses the sign of the determinant.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find the determinant of the first matrix

The determinant of the matrix \(\left[ {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right]\) can be calculated as shown below:

\(\left| {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right| = ad - bc\)

Step 2: Find the determinant of the second matrix

The determinant of the matrix \(\left[ {\begin{aligned}{*{20}{c}}c&d\\a&b\end{aligned}} \right]\) can be calculated as shown below:

\(\begin{aligned}{c}\left| {\begin{aligned}{*{20}{c}}c&d\\a&b\end{aligned}} \right| = bc - ad\\ = - \left( {ad - bd} \right)\end{aligned}\)

So, the row operation swaps row 1 and 2 of the matrix and reverses the sign of the determinant.

Most popular questions for Math Textbooks

Question: In Exercise 20, find the area of the parallelogram whose vertices are listed.

20. \(\left( {0,0} \right),\left( { - {\bf{2}},4} \right),\left( {4, - 5} \right),\left( {2, - 1} \right)\)

In Exercises 24–26, use determinants to decide if the set of vectors is linearly independent.

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

Question: In Exercise 19, find the area of the parallelogram whose vertices are listed.

19. \(\left( {0,0} \right),\left( {5,2} \right),\left( {6,4} \right),\left( {11,6} \right)\)

Find the determinants in Exercises 5-10 by row reduction to echelon form.

\(\left| {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{3}}&{ - {\bf{3}}}\\{\bf{3}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{ - {\bf{5}}}\end{aligned}} \right|\)

In Exercise 33-36, verify that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\)where E is the elementary matrix shown and \(A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\).

35. \(\left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\)

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