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Linear Algebra and its Applications
Found in: Page 165
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

Let \(A = \left[ {\begin{array}{*{20}{c}}3&1\\4&2\end{array}} \right]\). Write \(5A\). Is \(\det 5A = 5\det A\)?

\(\det 5A \ne 5\det A\)

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Determine the matrix \(5A\)

Let \(A = \left[ {\begin{array}{*{20}{c}}3&1\\4&2\end{array}} \right]\).

Compute the matrix \(5A\) as shown below:

\(\begin{array}{c}5A = 5\left[ {\begin{array}{*{20}{c}}3&1\\4&2\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{15}&5\\{20}&{10}\end{array}} \right]\end{array}\)

Step 2: Verify whether \(\det 5A = 5\det A\) 

The determinant of matrix A is shown below:

\(\begin{array}{c}\det A = \left| {\begin{array}{*{20}{c}}3&1\\4&2\end{array}} \right|\\ = 6 - 4\\ = 2\end{array}\)

The determinant of matrix \(5A\) is shown below:

\[\begin{array}{c}\det 5A = \left| {\begin{array}{*{20}{c}}{15}&5\\{20}&{10}\end{array}} \right|\\ = 150 - 100\\ = 50\end{array}\]

Thus, \(\det 5A \ne 5\det A\).

Most popular questions for Math Textbooks

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

\(\left| {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{3}}&{\bf{2}}&{ - {\bf{4}}}\\{\bf{0}}&{\bf{1}}&{\bf{2}}&{ - {\bf{5}}}\\{\bf{2}}&{\bf{7}}&{\bf{6}}&{ - {\bf{3}}}\\{ - {\bf{3}}}&{ - {\bf{10}}}&{ - {\bf{7}}}&{\bf{2}}\end{aligned}} \right|\)

Compute the determinant in Exercise 9 by cofactor expansions. At each step, choose a row or column that involves the least amount of computation.

9. \(\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{\bf{0}}&{\bf{5}}\\{\bf{1}}&{\bf{7}}&{\bf{2}}&{ - {\bf{5}}}\\{\bf{3}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\\{\bf{8}}&{\bf{3}}&{\bf{1}}&{\bf{7}}\end{array}} \right|\)

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

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

The expansion of a \({\bf{3}} \times {\bf{3}}\) determinant can be remembered by the following device. Write a second type of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals.

Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants in Exercises 15-18. Warning: This trick does not generalize in any reasonable way to \({\bf{4}} \times {\bf{4}}\) or larger matrices.

15. \(\left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{4}}\\{\bf{2}}&{\bf{3}}&{\bf{2}}\\{\bf{0}}&{\bf{5}}&{ - {\bf{2}}}\end{array}} \right|\)

Compute the determinants of the elementary matrices given in Exercise 25-30.

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