[Math] Estimations for the size of the biggest entry in an inverse Matrix

Question

If you got a Matrix $A$. Is there a estimation how big the largest element in the inverse of the matrix is?

If it helps the matrix is unimodular, that means all entries are integer and the determinant is +-1. The inverse is also unimodular.

Answer 1

Just for illustration how bad it can be: consider an $n\times n$ lower triangular matrix $A$ with $1$ on the diagonal and $-1$ in the strictly lower triangular part. The matrix is clearly unimodular. However, the entry $(n,1)$ of $A^{-1}$ is equal to $2^{n-2}$ (so it grows very fast with $n$ even though $A$ is full of quite "innocent" numbers).