If $P$ is an invertible transition probability matrix, does $P^{-1}[i,j]$ have any interesting meaning?

Question

Suppose we have a Markov chain transition probability matrix $P$ that is invertible, i.e., $P^{-1}$ exists.

Question: Does there exist a meaningful interpretation of the $(i,j)$ entry in $P^{-1}$?

So, for example, if $$P=\left(\begin{matrix} 0.3 & 0.7 \\ 0.1 & 0.9 \\ \end{matrix}\right)$$ then $$P^{-1}=\left(\begin{matrix} 4.5 & -3.5 \\ -0.5 & 1.5 \\ \end{matrix}\right).$$ Do these values, $4.5$, $-3.5$, etc., have any interesting meaning?

Answer 1

Since the inverse matrix may well contain negative terms, it is difficult to interpret it on its own. But we have

$$PP^{-1} = I = \Big[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \Big ] $$

Then $P^{-1}$ is the matrix that makes the transition matrix of the chain equal to $I$. Now, describe and interpret the situation of a markov chain having $I$ as its transition matrix.