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Chapter 3: Q27Q (page 165)
Compute the determinants of the elementary matrices given in Exercise 25-30.
27. \(\left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\k&0&1\end{array}} \right]\).
The determinant of the matrix is 1.
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Compute the determinants of the elementary matrices given in Exercise 25-30.
28. \(\left( {\begin{aligned}{*{20}{c}}k&0&0\\0&1&0\\0&0&1\end{aligned}} \right)\).
Compute the determinant in Exercise 6 using a cofactor expansion across the first row.
6. \(\left| {\begin{aligned}{*{20}{c}}{\bf{5}}&{ - {\bf{2}}}&{\bf{2}}\\{\bf{0}}&{\bf{3}}&{ - {\bf{3}}}\\{\bf{2}}&{ - {\bf{4}}}&{\bf{7}}\end{aligned}} \right|\)
Combine the methods of row reduction and cofactor expansion to compute the determinant in Exercise 11.
11. \(\left| {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{4}}&{ - {\bf{3}}}&{ - {\bf{1}}}\\{\bf{3}}&{\bf{0}}&{\bf{1}}&{ - {\bf{3}}}\\{ - {\bf{6}}}&{\bf{0}}&{ - {\bf{4}}}&{\bf{3}}\\{\bf{6}}&{\bf{8}}&{ - {\bf{4}}}&{ - {\bf{1}}}\end{aligned}} \right|\)
Question: Use Cramer鈥檚 rule to compute the solutions of the systems in Exercises1-6.
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.
\[\left| {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\\{\bf{7}}&{ - {\bf{2}}}&{\bf{0}}&{\bf{0}}\\{\bf{2}}&{\bf{6}}&{\bf{3}}&{\bf{0}}\\{\bf{3}}&{ - {\bf{8}}}&{\bf{4}}&{ - {\bf{3}}}\end{array}} \right|\]
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