Let \( A=\left[\begin{array}{ll}a & b \\ b & c\end{array}\right] \) where \( a, b, c \in Z \) and the sum of diagonal elements of \( A^{2} \) is 1 . If \( A_{1}, A_{2}, A_{3} \) and \( A_{4} \) are the distinct matrices satisfying the above condition then \( A_{1}+A_{2}+A_{3}+A_{4}= \)