Well it turns out its because both matrix and vector are stochastic. A stochastic matrix multiplied with a stochastic vector will always yield a stochastic vector (this can be shown) so by the rule of A(BC)=(AB)C the result will always have to be stochastic no matter how many times you take the matrix to a power and so the x and y "coordinates" must conform to x+y=1 which means a straight line.

Maybe this can help someone else.