Let A be a symmetric matrix of order 2 with integer entries. If the...

{A}^{2} = \left(\begin{cases}a & b \\ b & c\end{cases}\right)\left(\begin{cases}a & b \\ b & c\end{cases}\right) = \left(\begin{cases}{a}^{2} + {b}^{2} & b\left(a + c\right) \\ b\left(a + c\right) & {b}^{2} + {c}^{2}\end{cases}\right)

Sum of the diagonal entries of

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

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

$b = 0 \text{ and } {a}^{2} + {c}^{2} = 1$

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

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

Total $= 4$ matrices