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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.
23. \(\left( {\begin{aligned}{*{20}{c}}2&0&0&6\\1&{ - 7}&{ - 5}&0\\3&8&6&0\\0&7&5&4\end{aligned}} \right)\)
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.
\(\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{\bf{5}}&{\bf{4}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{5 + 3k}&{4 + 2k}\end{array}} \right]\)
Question: In Exercises 31–36, mention an appropriate theorem in your explanation.
33. Let A and B be square matrices. Show that even thoughABand BAmay not be equal, it is always true that\(det{\rm{ }}AB = det{\rm{ }}BA\).
Question: In Exercise 10, determine the values of the parameter s for which the system has a unique solution, and describe the solution.
10.
\(\begin{array}{c}s{x_{\bf{1}}} - {\bf{2}}{x_{\bf{2}}} = {\bf{1}}\\4s{x_{\bf{1}}} + {\bf{4}}s{x_{\bf{2}}} = {\bf{2}}\end{array}\)
Question: In Exercise 20, find the area of the parallelogram whose vertices are listed.
20. \(\left( {0,0} \right),\left( { - {\bf{2}},4} \right),\left( {4, - 5} \right),\left( {2, - 1} \right)\)
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