This query is inspired by this previous question.
Suppose $A$ is an $n \times n$ matrix whose entries are integers between $-s$ and $s$. Suppose further that $A^k=I$ and moreover $k$ is the smallest positive integer with this property. What sort of bounds can be derived on $k$ in terms of $n$ and $s$?
A related question is considered in the discussion to this answer. The question I am asking is slightly different because I am restricting the integers to lie between $-s$ and $s$.