Suppose that $A$ is an irreducible row stochastic matrix $n \times n$, and $mij$ are elements of the mean first passage matrix.
Deleting $j-th$ row and column of $A$ and calling it $A_{ij}$, is it a standard fact that $$m_{ij} = e_i^T (I-A_j)^{-1}1$$ Any help or hint would be appreciated.