Question

Determine the condition that the matrix
$A=\left[\begin{array}{cc}a& b\\ c& -a\end{array}\right]$ be involuntary i.e, $A^2=I$

Answer 1

$A=\left[\begin{array}{cc}a& b\\ c& -a\end{array}\right]$ is involuntary.

If a matrix is involuntary, then $A^2=I$

$A^2=A.A=\left[\begin{array}{cc}{a}^2+bc& 0\\ 0& a^2+bc\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\therefore a^2+bc=1$ is the required condition.