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Chapter 3: Q32Q (page 165)
Use Exercise 25-28 to answer the questions in Exercises 31 ad 32. Give reasons for your answers.
32. What is the determinant of an elementary scaling matrix with k on the diagonal?
The determinant of a \(3 \times 3\) elementary scaling matrix with \(k\) on the diagonal is \(k\).
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Find the determinant in Exercise 20, where \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right| = {\bf{7}}\].
20. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{{\bf{d}} + {\bf{3g}}}&{{\bf{e}} + {\bf{3h}}}&{{\bf{f}} + {\bf{3i}}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right|\]
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.
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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.
\(\left| {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{3}}&{\bf{4}}\\{\bf{2}}&{\bf{3}}&{\bf{1}}\\{\bf{3}}&{\bf{3}}&{\bf{2}}\end{aligned}} \right|\)
In Exercises 21鈥�23, use determinants to find out if the matrix is invertible.
22. \(\left( {\begin{aligned}{*{20}{c}}5&1&{ - 1}\\1&{ - 3}&{ - 2}\\0&5&3\end{aligned}} \right)\)
In Exercise 33-36, verify that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\)where E is the elementary matrix shown and \(A = \left[ {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right]\).
36. \(\left[ {\begin{aligned}{*{20}{c}}1&0\\0&k\end{aligned}} \right]\)
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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