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Chapter 3: Q11E (page 165)

Combine the methods of row reduction and cofactor expansion to compute the determinant in Exercise 11.

11. \(\left| {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{4}}&{ - {\bf{3}}}&{ - {\bf{1}}}\\{\bf{3}}&{\bf{0}}&{\bf{1}}&{ - {\bf{3}}}\\{ - {\bf{6}}}&{\bf{0}}&{ - {\bf{4}}}&{\bf{3}}\\{\bf{6}}&{\bf{8}}&{ - {\bf{4}}}&{ - {\bf{1}}}\end{aligned}} \right|\)

Short Answer

Expert verified

\(\left| {\begin{aligned}{*{20}{c}}3&4&{ - 3}&{ - 1}\\3&0&1&{ - 3}\\{ - 6}&0&{ - 4}&3\\6&8&{ - 4}&{ - 1}\end{aligned}} \right| = - 48\)

Step by step solution

01

Create zero in the second column

Add \( - 2\) times row 1 to row 4 to obtain

\(\left| {\begin{aligned}{*{20}{c}}3&4&{ - 3}&{ - 1}\\3&0&1&{ - 3}\\{ - 6}&0&{ - 4}&3\\6&8&{ - 4}&{ - 1}\end{aligned}} \right| \sim \left| {\begin{aligned}{*{20}{c}}3&4&{ - 3}&{ - 1}\\3&0&1&{ - 3}\\{ - 6}&0&{ - 4}&3\\0&0&2&1\end{aligned}} \right|\).

02

Use cofactor expansion down the second column

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

03

Create zero in the first column

Add 2 times row 1 to row 2 to obtain

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

04

Use cofactor expansion down the first column

\(\begin{aligned}{c} - 4\left| {\begin{aligned}{*{20}{c}}3&1&{ - 3}\\0&{ - 2}&{ - 3}\\0&2&1\end{aligned}} \right| = - 4\left( {3\left| {\begin{aligned}{*{20}{c}}{ - 2}&{ - 3}\\2&1\end{aligned}} \right| + 0 + 0} \right)\\ = - 4\left( {3\left( 4 \right)} \right)\\ = - 48\end{aligned}\)

Hence, \(\left| {\begin{aligned}{*{20}{c}}3&4&{ - 3}&{ - 1}\\3&0&1&{ - 3}\\{ - 6}&0&{ - 4}&3\\6&8&{ - 4}&{ - 1}\end{aligned}} \right| = - 48\).

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

25. \(\left( {\begin{aligned}{*{20}{c}}7\\{ - 4}\\{ - 6}\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{ - {\bf{8}}}\\{\bf{5}}\\7\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{\bf{7}}\\{\bf{0}}\\{ - {\bf{5}}}\end{aligned}} \right)\)

In Exercises 21鈥�23, use determinants to find out if the matrix is invertible.

21. \(\left( {\begin{aligned}{*{20}{c}}2&6&0\\1&3&2\\3&9&2\end{aligned}} \right)\)

Each equation in Exercises 1-4 illustrates a property of determinants. State the property

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

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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