Singular irreducible M-matrices revisited

There are several well-known necessary and sufficient conditions for an M-matrix to possess “property c”. Specializing some of these statements, the objective here is to characterize matrices that belong to the subclass of M-matrices with “property c”, namely, singular irreducible M-matrices. A new characterization for an M-matrix A to have “property c” is given in terms of a condition involving Z-matrices C such that A≤C. A result of Fan on M-matrices is improved, and extensions to M-matrices with “property c” and singular irreducible M-matrices, are obtained.