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Q.
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
$A=\begin{pmatrix}a&b\\ b&c\end{pmatrix}, a, b, c \in I $
$A^{2}=\begin{pmatrix}a&b\\ b&c\end{pmatrix}\begin{pmatrix}a&b\\ b&c\end{pmatrix}=\begin{pmatrix}a^{2}+b^{2}&b\left(a+c\right)\\ b\left(a+c\right)&b^{2}+c^{2}\end{pmatrix}$
Sum of the diagonal entries of
$A^{2}=a^{2}+2 b^{2}+c^{2}$
Given $a^{2}+2 b^{2}+c^{2}=1, a, b, c \in I$
$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