91Ó°ÊÓ

Compute the determinants of the elementary matrices given in Exercises 25-30. (See Section 2.2)

\[\left[ {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{1}}&{\bf{0}}\\{\bf{0}}&k&{\bf{1}}\end{aligned}} \right]\]

In Exercises 27 and 28, A and B are \[n \times n\] matrices. Mark each statement True or False. Justify each answer.

27. a. A row replacement operation does not affect the determinant of a matrix.

b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by \({\left( { - {\bf{1}}} \right)^r}\), where r is the number of row interchanges made during row reduction from A to U.

c. If the columns of A are linearly dependent, then \(det\left( A \right) = 0\).

d. \(det\left( {A + B} \right) = det{\rm{ }}A + det{\rm{ }}B\).

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?

Question: In Exercises 31–36, mention an appropriate theorem in your explanation.

34. Let A and P be square matrices, with P invertible. Show that \(det\left( {PA{P^{ - {\bf{1}}}}} \right) = det{\rm{ }}A\).

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. \(\left[ {\begin{aligned}{*{20}{c}}1&0\\0&k\end{aligned}} \right]\)

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 \(\det A\).

Question: Use Cramer’s rule to compute the solutions of the systems in Exercises1-6.

3. \(\begin{array}{c}3{x_1} - 2{x_2} = 3\\ - 4{x_1} + 6{x_2} = - 5\end{array}\)

Let \(u = \left[ {\begin{array}{*{20}{c}}3\\0\end{array}} \right]\), and \(v = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]\). Compute the area of the parallelogram

determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{array}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{array}} \right]\). How do they compare? Replace the first entry of v by an arbitrary number x, and repeat the problem. Draw a picture and explain what you find.

Find the determinants in Exercises 5-10 by row reduction to echelon form.

\(\left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{5}}&{ - {\bf{4}}}\\{ - {\bf{1}}}&{ - {\bf{4}}}&{\bf{5}}\\{ - {\bf{2}}}&{ - {\bf{8}}}&{\bf{7}}\end{array}} \right|\)