Here’s a really elementary proof (which is a slight modification of Fanfan’s answer to a question of mine). As Calle shows, it is easy to see that the eigenvalue 1 is obtained. Now, suppose Ax=λx for some λ>1. Since the rows of A are nonnegative and sum to 1, each element of vector Ax is a convex combination of the components of x, which can be no greater than xmax, the largest component of x. On the other hand, at least one element of λx is greater than xmax, which proves that λ>1 is impossible.