Let $A$ and $B$ be $n\times n$ matrices with integer entries. Show that if $B=A^{-1}$ then $A$ and $B$ are permutation matrices (matrix obtained by permuting rows of the identity matrix).
If the entries are non negative integers then it is quite easy to show, but if the entries also involve negative integers then I couldn't show it.
If the matrices may have negative integer entries, then the statement is not true. For example, $$A= \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix} $$ both have determinant $1$, and so are invertible over the integers, and in fact $AB = I$. But they are not permutation matrices.