A vector in R^n with nonnegative entries which add up to 1 is called probability vector. An n x n matrix whose columns are probability vectors is called stochastic matrix. Let A be such a stochastic matrix and let v be probability vector.
(a) Discuss that lambda = 1 is an eigenvalue of A.
(b) Let w = [1 1 ... 1]. Show that wv = 1
(c) Calculate wA.
(d) Discuss that Av is also a probability vector. Deduce that A^2 is also a stochastic matrix.