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Chapter 3: Q13E (page 165)

Combine the methods of row reduction and cofactor expansion to compute the determinant in Exercise 13.

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

Short Answer

Expert verified

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

Step by step solution

01

Create zero in the fourth column

Add \( - 2\) times row 1 to row 2 to obtain

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

02

Use cofactor expansion down the fourth column

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

03

Create zero in the first column

Add row 3 to row 2 to obtain

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

04

Use cofactor expansion down the first column

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

Hence, \(\left| {\begin{aligned}{*{20}{c}}2&5&4&1\\4&7&6&2\\6&{ - 2}&{ - 4}&0\\{ - 6}&7&7&0\end{aligned}} \right| = 6\).

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Most popular questions from this chapter

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.

\(\left[ {\begin{array}{*{20}{c}}a&b&c\\{\bf{3}}&{\bf{2}}&{\bf{1}}\\{\bf{4}}&{\bf{5}}&{\bf{6}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}&{\bf{1}}\\a&b&c\\{\bf{4}}&{\bf{5}}&{\bf{6}}\end{array}} \right]\)

Question: In Exercise 10, determine the values of the parameter s for which the system has a unique solution, and describe the solution.

10.

\(\begin{array}{c}s{x_{\bf{1}}} - {\bf{2}}{x_{\bf{2}}} = {\bf{1}}\\4s{x_{\bf{1}}} + {\bf{4}}s{x_{\bf{2}}} = {\bf{2}}\end{array}\)

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.

\(\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{\bf{5}}&{\bf{4}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{5 + 3k}&{4 + 2k}\end{array}} \right]\)

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.

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

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

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

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