Monotone Property of Matrix

Question

Let A be a monotone matrix i.e. $A^{-1}\geq 0$. We have if $Av\geq 0 \implies v\geq 0 \forall v \in \mathbb{R}^n$ Can we have the other side of implication? I tried but not able to get it.

Answer 1

No. E.g. $A=\pmatrix{1&-2\\ 0&1}$ is monotone, because $A^{-1}=\pmatrix{1&2\\ 0&1}\ge0$, but $A\pmatrix{0\\ 1}=\pmatrix{-2\\ 1}\not\ge0$.