  We introduce the concept of stable index for 0-1 matrices. Let $A$ be a 0-1
square matrix. If $A^k$ is a 0-1 matrix for every positive integer $k$, then
the stable index of $A$ is defined to be infinity; otherwise, the stable index
of $A$ is defined to be the smallest positive integer $k$ such that $A^{k+1}$
is not a 0-1 matrix. We determine the maximum finite stable index of all 0-1
matrices of order $n$ as well as the matrices attaining the maximum finite
stable index.
