Question

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

• A. Idempotent only
• B. Involutary only
• C. Involutary and Orthogonal
• D. Involutary and Idempotent

Answer 1

The correct option is B Involutary only

$A^2=A\times A=\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]\times \left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$
$=\left[\begin{array}{ccc}15-24+0& 40-40+0& 0+0+0\\ -15+15+0& -24+25+0& 0+0+0\\ -5+6-1& -8+10-2& 0+0+1\end{array}\right]$
$=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=I$
Hence, the given matrix $A$ is involutory.