Let me begin by admitting that I have no knowledge of the subject area in question: the following is my answer as a googlist, not a mathematician.

Having disclosed that, it seems that the result that you want can be found in Section 2.5 of *Topics in Matrix Analysis* by Horn and [C.R.] Johnson.

The extent of my grasp of the material at the moment is the following: an inverse $M$-matrix is an invertible matrix whose inverse is an $M$-matrix.

**Addendum**: Caveat lector: I looked more closely at the section in question and found the following passage (starting at the bottom of p. 119):

"Exercise. Show that: a) The principal minors of an inverse $M$-matrix are positive; b) every principal submatrix of an inverse $M$-matrix is an inverse $M$-matrix; and c)...Verification of these facts requires some effort."

So one sees the limits of the approach of googling and then quoting from texts: sometimes one has to put some thought into the matter! Probably someone else will enjoy reading this section of the book and working out the exercise, so I'll leave this response up, although it is certainly not an answer in and of itself.