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Bipartite graphs are used todescribe the generalized Schur complements of real matrices having nosquare submatrix with twoor more nonzerodiagonals. For any matrix A with this property, including any nearly reducible matrix, the sign pattern of each generalized Schur complement is shown to be determined uniquely by the sign pattern of A. Moreover, if A has a normalized LU factorization A = LU, then the sign pattern of A is shown to determine uniquely the sign patterns of L and U, and (with the standard LU factorization) of L−1 and, if A is nonsingular, of U−1. However, if A is singular, then the sign pattern of the Moore-Penrose inverse U† may not be uniquely determined by the sign pattern of A. Analogous results are shown to hold for zero patterns.