This problem tests your understanding of matrix algebra, specifically the binomial expansion for matrices and the properties of idempotent matrices. We are given a condition involving a matrix $A$ and need to find the number of such matrices whose elements are restricted to $\{-1, 0, 1\}$.

1. Key Concept: Binomial Expansion for Matrices

For any two matrices $X$ and $Y$ that commute (i.e., $XY = YX$), the binomial expansion formula holds: $(X \pm Y)^n = \sum_{k=0}^n \binom{n}{k} X^{n-k} (\pm Y)^k$ In our case, we have $(I - A)^3$. Since the identity matrix $I$ commutes with any