Let $A$ be an $N\times N$ strictly substochastic matrix (i.e., $a_{ij}\in[0,1)$ and $\sum_{j}a_{ij}<1$), let $B=I-A$, and let $\tilde{B}$ be the matrix obtained from $B$ by removing the first row and the first column. Finally, let $M$ be the adjugate matrix of $B$ and $\tilde{M}$ be the adjugate matrix of $\tilde{A}$. For $N=3$ below I show that $$\frac{\tilde{M}_{22}}{\mathrm{det}\tilde{B}}\leq\frac{M_{22}}{\det B}.$$ Note that this implies that the $(2,2)$ entry of $\tilde{B}^{-1}$ is weakly lower than the corresponding entry of $B^{-1}$. Does this result generalize to $N>3$? In the proof below I only use that $B$ has positive diagonal and negative off-diagonal terms, so I think this is a property of all M-matrices.
In the case in which $N=3$ then $$ \tilde{M}_{22}=1-a_{33} $$ $$ \det\tilde{B}=\left(1-a_{22}\right)\left(1-a_{33}\right)-a_{23}a_{32} $$ $$ M_{22}=\left(1-a_{11}\right)\left(1-a_{33}\right)-a_{13}a_{31} $$ $$ \det B=-a_{31}\left(a_{12}a_{23}+a_{13}\left(1-a_{22}\right)\right)+a_{32}\left(-\left(1-a_{11}\right)a_{23}-a_{13}a_{21}\right)+\left(1-a_{33}\right)\left(\left(1-a_{11}\right)\left(1-a_{22}\right)-a_{12}a_{21}\right) $$ so we need to show that \begin{align*} -a_{31}\left(1-a_{33}\right)\left(a_{12}a_{23}+a_{13}\left(1-a_{22}\right)\right)+a_{32}\left(1-a_{33}\right)\left(-\left(1-a_{11}\right)a_{23}-a_{13}a_{21}\right)+\left(1-a_{33}\right)^{2}\left(\left(1-a_{11}\right)\left(1-a_{22}\right)-a_{12}a_{21}\right)\\ \leq\left(\left(1-a_{22}\right)\left(1-a_{33}\right)-a_{23}a_{32}\right)\left(\left(1-a_{11}\right)\left(1-a_{33}\right)-a_{13}a_{31}\right) \end{align*} Eliminating matching terms on both sides leads to $$ -a_{12}a_{23}a_{31}\left(1-a_{33}\right)-a_{13}a_{21}a_{32}\left(1-a_{33}\right)-a_{12}a_{21}\left(1-a_{33}\right)^{2}\leq-a_{13}a_{23}a_{31}a_{32} $$ which obviously holds.
How do I extend this proof for $N>3$? One possibility I see is to use this result here: https://math.stackexchange.com/a/1250091/165163
