I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in Explicit formula for Cholesky factorization in a special case, but the question might have general interest). This matrix is not a $Z$-matrix so the whole body of knowledge that had been developed for them is not suitable.

What I'd like to find is a simple sufficient condition that I can easily analyze. The best approximation to that that I've found is Peris's criterion using $B$-splittings but it still didn't help me enough.

Are you aware of any such results?