# Transition matrix inverse

If you have a Markov chain, such that

$x_i A = x_{i+1}$,

is it true that:

$x_i = x_{i+1} A^{-1}$

Further, in the strange case where $A$ may be singular, is it appropriate to compute the Moore-Penrose pseudo-inverse to get an approximate answer?

Thanks!

• Sure, just multiply both sides by $A^{-1}$. – Qiaochu Yuan Aug 10 '17 at 3:07

• Analogous: I roll a fair die 10 times independently. What info about the 9th roll can be gained from the 10th? After some thought, I guess there is one less interesting kind of transition matrix from a probability point of view: the $n \times n$ identity (1's on the diagonal); a chain with no movement at all. – BruceET Aug 10 '17 at 18:59