Recall that if M is an n n matrix whose entries, mij , are non-negative, i.e., 0 mij then M is called a stochastic matrix if for each j = 1, 2, . . . , n n i mij = 1. Also recall that an n 1 vector X with non-negative entires, xi is called a state vector if n i xi = 1. (a) Show that if M is a stochastic matrix and X is a state vector, then Y = M X is also a state vector. (b) Show that if M is a stochastic matrix, the so is M k for any positive integer k. (c) Given an initial state vector X0 define recursively a sequence of vectors, Xk 1 = M Xk, k = 0, 1, . . .. Show that Xk = M kX0