Consider a basic Markov process where vector $x$ at time $t$ switches states with a transition matrix $A$, so that: $$ x(t+1) = A x(t). $$
Assume that A is invertible, we can write the above as: $$ A^{-1}x(t+1) = x(t) $$
What is the meaning of $A^{-1}$? Does it have any special properties?