Question

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

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 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 \( - 1\).