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Chapter 3: Q30Q (page 165)

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

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

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 cofactor expansion across any row or down any column. Expansion across the

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

Cofactor expansion across row 1 is shown below:

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

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

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Most popular questions from this chapter

Compute \(det{\rm{ }}{B^4}\), where \(B = \left[ {\begin{aligned}{{}{}}1&0&1\\1&1&2\\1&2&1\end{aligned}} \right]\).

Combine the methods of row reduction and cofactor expansion to compute the determinant in Exercise 13.

13. \(\left| {\begin{aligned}{*{20}{c}}{\bf{2}}&{\bf{5}}&{\bf{4}}&{\bf{1}}\\{\bf{4}}&{\bf{7}}&{\bf{6}}&{\bf{2}}\\{\bf{6}}&{ - {\bf{2}}}&{ - {\bf{4}}}&{\bf{0}}\\{ - {\bf{6}}}&{\bf{7}}&{\bf{7}}&{\bf{0}}\end{aligned}} \right|\)

In Exercises 21鈥�23, use determinants to find out if the matrix is invertible.

21. \(\left( {\begin{aligned}{*{20}{c}}2&6&0\\1&3&2\\3&9&2\end{aligned}} \right)\)

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 \(\det A\).

In Exercise 33-36, verify that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\)where E is the elementary matrix shown and \(A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\).

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