1. The problem statement, all variables and given/known data Let P be a stochastic matrix on a finite set I. Show that a distribution π is invariant for P if and only if π(I-P+A) = a, where A = (aij : i,j in I) with aij = 1 for all i and j, and a = (ai : i in I) with ai = 1 for all i. Deduce that if P is irreducible then I-P+A is invertible. Note that this enables one to compute the invariant distribution by any standard method of inverting a matrix. 2. Relevant equations 3. The attempt at a solution I have no idea how I approach to the answer. Anybody give me some idea.