Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A 2 is 1…

$A^2=\left(\begin{array}{ll}a& b\\ b& c\end{array}\right)\left(\begin{array}{ll}a& b\\ b& c\end{array}\right)=\left(\begin{array}{cc}{a}^2+b^2& b\left(a+c\right)\\ b\left(a+c\right)& b^2+c^2\end{array}\right)$

Sum of the diagonal entries of

$A^2=a^2+2b^2+c^2$

Given $a^2+2b^2+c^2=1,a,b,c\in I$

$b=0 \& a^2+c^2=1$

Case-1: $a=0\Rightarrow c=\pm 1$   ($2$-matrices)

Case-2: $c=0\Rightarrow a=\pm 1$   ($2$-matrices)

Total $=4$ matrices