Question

If $A=\left[\begin{array}{cc}a& -1\\ 15& -4\end{array}\right]$ is an involutory matrix, then the value of $a$ is

Answer 1

Given $A=\left[\begin{array}{cc}a& -1\\ 15& -4\end{array}\right]$ is an involutory matrix.
We know that for an involutory matrix, $A^2=I$
$A^2=\left[\begin{array}{cc}a& -1\\ 15& -4\end{array}\right]\left[\begin{array}{cc}a& -1\\ 15& -4\end{array}\right]$
$=\left[\begin{array}{cc}{a}^2-15& -a+4\\ 15a-60& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
Equating the terms, we get
$a^2-15=1\Rightarrow a=\pm 4,$
$-a+4=0\Rightarrow a=4,$
$15a-60=0\Rightarrow a=4$
$\therefore a=4$