Let A, B be real square matrices such that A ≤ B, where the order is defined entrywise. Consider the interval J([A, B]) := {X : A ≤ X ≤ B}. In general, if A and B are invertible then there may exist X ∈ J([A, B]) which is not invertible. Consider now the case when A and B are invertible and that their inverses are (entry-wise) nonnegative. We ask if each X ∈ J ([A, B]) inherits this property that is if each X ∈ J ([A, B]) is invertible and its inverse is nonnegative. An answer is given. A similar question is posed for the nonnegativity of the Moore-Penrose inverse. The objective of this talk is to present some recent results in this direction.