Suppose $A$ is an $n\times n$ stochastic matrix, that is, entrywise nonnegative and row sums are all $1$. If $A$ is invertible, is it true that the minimum diagonal entry of $A^{-1}$ is no larger than $1$?

Small matrices support this claim, but for larger ones, I don't know how to (dis)prove it.

**Edited** I forgot to add the condition that the diagonal entries of $A$ are all zero.