Number of nonzero entries of S2NS matrices and matrices with signed generalized inverses

A real matrix A has a signed generalized inverse (or signed GI), if the sign pattern of its generalized inverse A+ is uniquely determined by the sign pattern of A. The notion of matrices having signed GI’s is a generalization of the well known notion of strong SNS matrices (or S2NS matrices). Sharp bounds, and characterization of equality, for the number of nonzero entries of S2NS matrices of order n are given. Then sharp bounds, and characterization of equality, for the number of nonzero entries of m×n matrices with signed GI’s are given