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Chapter 3: Q37Q (page 165)

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

Short Answer

Expert verified

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

Step by step solution

01

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:

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

02

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

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Most popular questions from this chapter

Compute the determinants in Exercises 9-14 by cofactor expnasions. At each step, choose a row or column that involves the least amount of computation.

\(\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{ - {\bf{7}}}&{\bf{3}}&{ - {\bf{5}}}\\{\bf{0}}&{\bf{0}}&{\bf{2}}&{\bf{0}}&{\bf{0}}\\{\bf{7}}&{\bf{3}}&{ - {\bf{6}}}&{\bf{4}}&{ - {\bf{8}}}\\{\bf{5}}&{\bf{0}}&{\bf{5}}&{\bf{2}}&{ - {\bf{3}}}\\{\bf{0}}&{\bf{0}}&{\bf{9}}&{ - {\bf{1}}}&{\bf{2}}\end{array}} \right|\)

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.

\(\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right],\left[ {\begin{array}{*{20}{c}}a&b\\{kc}&{kd}\end{array}} \right]\)

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

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

Compute the determinant in Exercise 4 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.

4. \(\left| {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{2}}&{\bf{4}}\\{\bf{3}}&{\bf{1}}&{\bf{1}}\\{\bf{2}}&{\bf{4}}&{\bf{2}}\end{aligned}} \right|\)

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