I want to show that an irreducibly diagonally dominant matrix with non-positive off-diagonal elements has a non-negative inverse.
I already know how to prove an irreducible diagonally dominant matrix is invertible. However, I have no idea about the proof of the positivity of the inverse.
My attempt: $A$ is invertible and $A^{-1}$ is nonnegative $\Longleftrightarrow $ If $Ax\ge Ay,$ then $x\ge y$ $\Longleftrightarrow$ If $Ax\ge 0,$ then $x\ge 0$
