Question

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

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

Step-by-step solution

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 \( - 1\).