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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

Use Exercise 25-28 to answer the questions in Exercises 31 ad 32. Give reasons for your answers.

32. What is the determinant of an elementary scaling matrix with k on the diagonal?

Question: 6. Use Cramer鈥檚 rule to compute the solution of the following system.

\(\begin{array}{c}{x_{\bf{1}}} + {\bf{3}}{x_{\bf{2}}} + \,{x_{\bf{3}}} = {\bf{4}}\\ - {x_{\bf{1}}} + \,\,\,\,\,\,\,\,\,\,{\bf{2}}{x_{\bf{3}}} = {\bf{2}}\\{\bf{3}}{x_{\bf{1}}} + \,{x_{\bf{2}}}\,\,\,\,\,\,\,\,\,\,\,\,\, = {\bf{2}}\end{array}\)

Compute the determinant in Exercise 8 using a cofactor expansion across the first row.

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

Question: In Exercise 16, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.

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

Find the determinant in Exercise 20, where \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right| = {\bf{7}}\].

20. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{{\bf{d}} + {\bf{3g}}}&{{\bf{e}} + {\bf{3h}}}&{{\bf{f}} + {\bf{3i}}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right|\]
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