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Chapter 3: Q13SE (page 165)
Question: 13. Show that if A is invertible, then adj A is invertible, and \({\left( {adj\,A} \right)^{ - {\bf{1}}}} = \frac{{\bf{1}}}{{detA}}A\).
Hence, adj Ais invertible and \({\left( {{\rm{adj}}\,A} \right)^{ - 1}} = \frac{1}{{\det A}}A\).
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Question: In Exercise 20, find the area of the parallelogram whose vertices are listed.
20. \(\left( {0,0} \right),\left( { - {\bf{2}},4} \right),\left( {4, - 5} \right),\left( {2, - 1} \right)\)
Find the determinant in Exercise 18, where \(\left| {\begin{aligned}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right| = {\bf{7}}\).
18. \(\left| {\begin{aligned}{*{20}{c}}{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right|\)
Question: In Exercise 8, determine the values of the parameter s for which the system has a unique solution, and describe the solution.
8.
\(\begin{array}{c}{\bf{3}}s{x_{\bf{1}}} + {\bf{5}}{x_{\bf{2}}} = {\bf{3}}\\12{x_{\bf{1}}} + {\bf{5}}s{x_{\bf{2}}} = {\bf{2}}\end{array}\)
In Exercises 39 and 40, \(A\) is an \(n \times n\) matrix. Mark each statement True or False. Justify each answer.
39.
a. An \(n \times n\) determinant is defined by determinants of \(\left( {n - 1} \right) \times \left( {n - 1} \right)\) submatrices.
b. The \(\left( {i,j} \right)\)-cofactor of a matrix \(A\) is the matrix \({A_{ij}}\) obtained by deleting from A its \(i{\mathop{\rm th}\nolimits} \) row and \[j{\mathop{\rm th}\nolimits} \]column.
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.
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