Nonnegative rectangular matrices having nonnegative Unweighted group inverses are characterized. Our techniques suggest an interesting approach to extend the earlier known results on X-monotone square matrices to rectangular ones. We also answer a question of characterizing nonnegative matrices having a nonnegative solution Y where (1) A = AXA, (2) X = XAX, (3) (AX) is O-symmetric, (4) ( XA) is...