Question

Show that $A=\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$ is involutory.

Answer 1

A square matrix $A$ is called involuntory provided it satisfies the relation

$A^2=I$, where $I$ is the identity matrix.

$A=\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$

$A^2=\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$

$=\left[\begin{array}{ccc}25-24+0& 40-40+0& 0+0+0\\ -15+15+0& -24+25+0& 0+0+0\\ -5+6-1& -8+10-2& 0+0+1\end{array}\right]$

$=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=I$

Hence the given matrix $A$ is involuntory.