I am trying to figure out the structure of an M-matrix (https://en.wikipedia.org/wiki/M-matrix) whose inverse has a special form: Let $A$ be an inverse M-matrix (inverse M-matrices are those matrices whose inverse is an M-matrix, such that each row sum of the matrix is a fixed constant (greater than 1). Each diagonal entry is strictly greater than all off-diagonal entries of that row such that $xa_{ii}\leq a_{ij}\leq ya_{ii}$ $\forall i\neq j$, where $0<x<y<1$. I am trying to show that $A^{-1}$ will have positive diagonal entries less than 1.

So far, I could not find any reference on why such a result must hold true, but I could not get a counter-example either (I tried numerical examples). Intuitively, I think it will be true due to the special structure of the matrix, the off-diagonal entries of $A$ are close to each other, so the inverse will have the above structure to compensate for that. Any idea or suggestion will be really helpful.