Show that when x is a probability vector and A is a stochastic matrix, then Ax is another probability matrix
clearly each coordinate of Ax is non-negative being the sum of products of non-negative numbers.
let us call the i,j-th entry of A, a_{ij}.
then (Ax)_{i} = a_{i1}x_{1} +...+ a_{in}x_{n}.
hence (Ax)_{1} +...(Ax)_{i} +...+ (Ax)_{n} =
(a_{11}x_{1} +...+ a_{1n}x_{n}) +...+ (a_{i1}x_{1} +...+ a_{in}x_{n}) +...+ (a_{n1}x_{1}+ ...+ a_{nn}x_{n}) =
(a_{11} +...+ a_{n1})x_{1} +...+ (a_{1j} +...+ a_{nj})x_{j} +...+ (a_{n1} +...+ a_{nn})x_{n}.
but a_{1j} +...+ a_{nj} = 1, since A is a stochastic matrix (this is just the sum of the j-th column, which is a probability vector), for EACH j,
hence (Ax)_{1} +...+ (Ax)_{n} = (a_{11} +...+ a_{n1})x_{1} +....+ (a_{n1} +...+ a_{nn})x_{n} = x_{1} +... + x_{n} = 1, so Ax is a probability vector.