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Chapter 3: Q29Q (page 165)
Compute the determinants of the elementary matrices given in Exercise 25-30.
29. \(\left[ {\begin{aligned}{*{20}{c}}1&0&0\\0&k&0\\0&0&1\end{aligned}} \right]\).
The determinant of the matrix is \(k\).
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Use Theorem 3 (but not Theorem 4) to show that if two rows of a square matrix A are equal, then \(det A = 0\). The same is true for twocolumns. Why?
Question: 13. Show that if A is invertible, then adj A is invertible, and \({\left( {adj\,A} \right)^{ - {\bf{1}}}} = \frac{{\bf{1}}}{{detA}}A\).
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.
\(\left[ {\begin{array}{*{20}{c}}a&b&c\\{\bf{3}}&{\bf{2}}&{\bf{1}}\\{\bf{4}}&{\bf{5}}&{\bf{6}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}&{\bf{1}}\\a&b&c\\{\bf{4}}&{\bf{5}}&{\bf{6}}\end{array}} \right]\)
Find the determinants in Exercises 5-10 by row reduction to echelon form.
\(\left| {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{3}}&{\bf{2}}&{ - {\bf{4}}}\\{\bf{0}}&{\bf{1}}&{\bf{2}}&{ - {\bf{5}}}\\{\bf{2}}&{\bf{7}}&{\bf{6}}&{ - {\bf{3}}}\\{ - {\bf{3}}}&{ - {\bf{10}}}&{ - {\bf{7}}}&{\bf{2}}\end{aligned}} \right|\)
In Exercise 33-36, verify that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\)where E is the elementary matrix shown and \(A = \left[ {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right]\).
33. \(\left[ {\begin{aligned}{*{20}{c}}1&k\\0&1\end{aligned}} \right]\)
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