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Chapter 3: Q27Q (page 165)

Compute the determinants of the elementary matrices given in Exercise 25-30.

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

Short Answer

Expert verified

The determinant of the matrix is 1.

Step by step solution

01

Compute the determinant of the elementary matrix

IfA is a triangular matrix, then according to theorem 2,det A is the product of the entries on its main diagonal.

The determinant of the matrix is the product of the diagonal entries because the matrix is triangular.

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\k&0&1\end{array}} \right| = \left( 1 \right)\left( 1 \right)\left( 1 \right)\\ = 1\end{array}\)

Thus, the determinant of the matrix is 1.

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Most popular questions from this chapter

Question: In Exercise 14, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.

14. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{1}}}&{\bf{2}}\\{\bf{0}}&{\bf{2}}&{\bf{1}}\\{\bf{2}}&{\bf{0}}&{\bf{4}}\end{array}} \right)\)

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

5. \(\begin{array}{c}{x_1} + {x_2} = 3\\ - 3{x_1} + 2{x_3} = 0\\{x_2} - 2{x_3} = 2\end{array}\)

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

\(\left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{3}}&{ - {\bf{1}}}&{\bf{0}}&{ - {\bf{2}}}\\{\bf{0}}&{\bf{2}}&{ - {\bf{4}}}&{ - {\bf{2}}}&{ - {\bf{6}}}\\{ - {\bf{2}}}&{ - {\bf{6}}}&{\bf{2}}&{\bf{3}}&{{\bf{10}}}\\{\bf{1}}&{\bf{5}}&{ - {\bf{6}}}&{\bf{2}}&{ - {\bf{3}}}\\{\bf{0}}&{\bf{2}}&{ - {\bf{4}}}&{\bf{5}}&{\bf{9}}\end{array}} \right|\)

Compute the determinant in Exercise 2 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.

  1. \(\left| {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{4}}&{\bf{1}}\\{\bf{5}}&{ - {\bf{3}}}&{\bf{0}}\\{\bf{2}}&{\bf{3}}&{\bf{1}}\end{aligned}} \right|\)
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