Inverse invariant zero–nonzero patterns

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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.