Inverse M matrices sufficient condition for reversing inequality

Question

I am asking out of curiosity, from a related question: Element-wise ordering of the inverse of two M-matrices.
I know that in general the converse is not true.
But suppose that we are given an inverse M matrix $A$ and identity matrix $I$ with $A \leq I$ (entrywise ordering), when can we say $A^{-1}\geq I$ holds true?
Try: Because $A$ is an inverse M matrix, I tried by assuming, suppose $A^{-1}$ is strictly diagonally dominant, then manipulate to show that the inequality will reverse, but so far no luck. Any hint or idea to try pursue will be really helpful.

Answer 1

If $A\le I$, $A$ is a diagonal matrix. Since $A$ is the inverse of an $M$-matrix, it is a positive diagonal matrix. The assumption $A\le I$ thus implies that all diagonal entries of $A$ lie inside $(0,1]$. Hence $A^{-1}\ge I$.