Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $A^2$ is 1, then the possible number of such matrices is :

Let  $A=\left[\begin{array}{l}ab\\ bc\end{array}\right]$and $A^2=\left[\begin{array}{l}ab\\ bc\end{array}\right]\left[\begin{array}{l}ab\\ bc\end{array}\right]=\left[\begin{array}{l}{a}^2+b^{2}ab+bc\\ ab+bc{b}^2+c^2\end{array}\right]$ 

            Given $a^2+2b^2+c^2=1$  and $a,b,c$ are integers. 

            The possible values are  $a=\pm 1,b=c=0$(or) ,$a=b=0$ and $c=\pm 1$

            Therefore, the number of possible matrices is 4