Chapter 3: Q37Q (page 165)
Let \(A = \left[ {\begin{array}{*{20}{c}}3&1\\4&2\end{array}} \right]\). Write \(5A\). Is \(\det 5A = 5\det A\)?
\(\det 5A \ne 5\det A\)
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In Exercises 21–23, use determinants to find out if the matrix is invertible.
21. \(\left( {\begin{aligned}{*{20}{c}}2&6&0\\1&3&2\\3&9&2\end{aligned}} \right)\)
In Exercises 24–26, use determinants to decide if the set of vectors is linearly independent.
24. \(\left( {\begin{aligned}{*{20}{c}}4\\6\\2\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{ - 7}\\0\\7\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{ - 3}\\{ - 5}\\{ - 2}\end{aligned}} \right)\)
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?
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 1 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.
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