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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Compute the determinants in Exercises 9-14 by cofactor expansions. At each step, choose a row or column that involves the least amount of computation.
\(\left| {\begin{aligned}{*{20}{c}}{\bf{6}}&{\bf{3}}&{\bf{2}}&{\bf{4}}&{\bf{0}}\\{\bf{9}}&{\bf{0}}&{ - {\bf{4}}}&{\bf{1}}&{\bf{0}}\\{\bf{8}}&{ - {\bf{5}}}&{\bf{6}}&{\bf{7}}&{\bf{1}}\\{\bf{2}}&{\bf{0}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\\{\bf{4}}&{\bf{2}}&{\bf{3}}&{\bf{2}}&{\bf{0}}\end{aligned}} \right|\)
Question: In Exercise 13, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.
13. \(\left( {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{5}}&{\bf{4}}\\{\bf{1}}&{\bf{0}}&{\bf{1}}\\{\bf{2}}&{\bf{1}}&{\bf{1}}\end{array}} \right)\)
Construct a random \({\bf{4}} \times {\bf{4}}\) matrix A with integer entries between \( - {\bf{9}}\) and 9. How is \(det {A^{ - 1}}\) related to \(det A\)? Experiment with random \({\bf{n}} \times {\bf{n}}\) integer matrices for \(n = 4\), 5, and 6, and make a conjecture. Note:In the unlikely event that you encounter a matrix with a zero determinant, reduce
it to echelon form and discuss what you find.
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. \(\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{\bf{0}}&{\bf{5}}\\{\bf{1}}&{\bf{7}}&{\bf{2}}&{ - {\bf{5}}}\\{\bf{3}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\\{\bf{8}}&{\bf{3}}&{\bf{1}}&{\bf{7}}\end{array}} \right|\)
In Exercises 31-36, mention an appropriate theorem in your explanation.
36. Find a formula for \(\det \left( {rA} \right)\) when \(A\) is an\(n \times n\) matrix.
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