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Chapter 3: Q25E (page 165)
Compute the determinants of the elementary matrices given in Exercises 25-30. (See Section 2.2)
\[\left[ {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{1}}&{\bf{0}}\\{\bf{0}}&k&{\bf{1}}\end{aligned}} \right]\]
The value of the determinant is 1.
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Let \(A = \left[ {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right]\) and let \(k\) be a scalar. Find a formula that relates \(\det kA\) to \(k\) and \(\det A\).
Construct a random \({\bf{4}} \times {\bf{4}}\) matrix A with integer entries between \( - {\bf{9}}\) and 9, and compare det A with det\({A^T}\), \(det\left( { - A} \right)\), \(det\left( {{\bf{2}}A} \right)\), and \(det\left( {{\bf{10}}A} \right)\). Repeat with two other random \({\bf{4}} \times {\bf{4}}\) integer matrices, and make conjectures about how these determinants are related. (Refer to Exercise 36 in Section 2.1.) Then check your conjectures with several random \({\bf{5}} \times {\bf{5}}\) and \({\bf{6}} \times {\bf{6}}\) integer matrices. Modify your conjectures, if necessary, and report your results.
Question: 13. Show that if A is invertible, then adj A is invertible, and \({\left( {adj\,A} \right)^{ - {\bf{1}}}} = \frac{{\bf{1}}}{{detA}}A\).
Question: Use Cramer鈥檚 rule to compute the solutions of the systems in Exercises1-6.
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.
\(\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right],\left[ {\begin{array}{*{20}{c}}a&b\\{kc}&{kd}\end{array}} \right]\)
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