Inverse invariant zero-nonzero patterns

We determine the irreducible zero-nonzero patterns A such that for any nonsingular matrix B over a field with zero-nonzero pattern A, the inverse B−1 has the same zero-nonzero pattern A. We also determine the zero-nonzero patterns P such that for any nonsingular matrix Q over a field with zero-nonzero pattern P, the transpose of the inverse (Q−1)T has the same zero-nonzero pattern P. One application of these results is to deduce the corresponding results on sign patterns.