Symmetric Matrix

$A=\left[\begin{array}{cc}\hfill p\hfill & \hfill q\hfill \\ \hfill r\hfill & \hfill s\hfill \end{array}\right]$ is symmetric. So, q = r.

$AA=A^2=\left[\begin{array}{cc}\hfill p\hfill & \hfill q\hfill \\ \hfill r\hfill & \hfill s\hfill \end{array}\right]\left[\begin{array}{cc}\hfill p\hfill & \hfill q\hfill \\ \hfill r\hfill & \hfill s\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill p^2+qr\hfill & \hfill pq+qs\hfill \\ \hfill rp+rs\hfill & \hfill rq+s^2\hfill \end{array}\right]$

Sum of diagonal elements =

$p^2+qr+rq+s^2=1\Rightarrow p^2+2r^2+s^2=1,\left(asq=r\right),p=0,r=0ands=\pm 1orr=0,s=0andp=\pm 1$

Total number of matrices = 4.