On the context of irreducible Markov chains on a finite graph one defines
$$G = \sum_{k=0}^\infty \hat{Q}^k $$
where $\hat{Q}$ is the restriction of a stochastic matrix to a subset (one can think that $\hat{Q}$ is equal to $Q$ after we eliminate some row $1$ and line $1$.)
this $\hat{Q}$ is a substochastic matrix and the above sum converges (see for instance Perron-Frobenius' theorem)
Now we note that $G = (I - \hat{G})^{-1}$ and that $G$ is positive definite.
then one considers the restriction of $G$ to a subset $A$ of the graph.
the question is
why is $G_A$ (the restriction of $G$ to a subset $A$ of the graph) non-degenerate? That is, why is $G_A$ invertible?