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Chapter 3: Q26Q (page 165)

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

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

Short Answer

Expert verified

The determinant of the matrix is \( - 1\).

Step by step solution

01

Compute the determinant of the elementary matrices

Theorem 1states that the determinant of an \(n \times n\) matrix A can be computed by a cofactor expansion across any row or down any column. The expansion across the

\(i{\mathop{\rm th}\nolimits} \)row using cofactors in \({C_{ij}} = {\left( { - 1} \right)^{i + j}}\det \,{A_{ij}}\) gives \(\det \,A = {a_{i1}}{C_{i1}} + {a_{i2}}{C_{i2}} + ... + {a_{in}}{C_{in}}\).

The cofactor expansion across row 1 is shown below:

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

Thus, the determinant of the matrix is \( - 1\).

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Most popular questions from this chapter

Compute the determinants in Exercises 9-14 by cofactor expnasions. At each step, choose a row or column that involves the least amount of computation.

\(\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{ - {\bf{7}}}&{\bf{3}}&{ - {\bf{5}}}\\{\bf{0}}&{\bf{0}}&{\bf{2}}&{\bf{0}}&{\bf{0}}\\{\bf{7}}&{\bf{3}}&{ - {\bf{6}}}&{\bf{4}}&{ - {\bf{8}}}\\{\bf{5}}&{\bf{0}}&{\bf{5}}&{\bf{2}}&{ - {\bf{3}}}\\{\bf{0}}&{\bf{0}}&{\bf{9}}&{ - {\bf{1}}}&{\bf{2}}\end{array}} \right|\)

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.

\(\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{\bf{5}}&{\bf{4}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{5 + 3k}&{4 + 2k}\end{array}} \right]\)

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. \(\left( {\begin{aligned}{*{20}{c}}4\\6\\2\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{ - 7}\\0\\7\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{ - 3}\\{ - 5}\\{ - 2}\end{aligned}} \right)\)

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

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