Let
$$A = \begin{bmatrix} A_1 & 0 & ... & 0 \\ 0 & A_2 & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & ... & A_n \end{bmatrix}$$
be a block diagonal matrix where $A_i$ is negative definite. $B$ is a square matrix whose elements are bounded between two positive numbers such that $0<\underline{B}\leq B_{ij}\leq \bar{B}$ for all $i,j$.
I would like to find conditions on the elements of $A$ such that all the elements of the matrix $$(A-B)^{-1}$$ are nonnegative.
One idea is to impose conditions such that $A-B$ is an M-matrix (such that all its off-diagonal terms are nonpositive and the real part of its eigenvalues are nonnegative). In this case, we know that $A-B$ is invertible and $(A-B)^{-1}\geq 0$. But I struggle to go from there to conditions on the elements $A_{ij}$.