How do we prove that Stochastic matrices preserve l1 norm?

Question

I have tried some previously answered questions on the forum but I don't see a formal proof anywhere. I can't understand where to begin either.

Essentially given that $A$ is a stochastic matrix and $b$ is a vector, I need to prove that $\|Ab\|_1 = \|b\|_1$

Answer 1

This is not true. $A =\begin{bmatrix} 1/2 \,\,1/2\,\\ 0\,\,\,\,\,\, 1\end{bmatrix}$ and $b=(0,1)^{T}$ is a counter-example. Here $Ab=(1/2,1)^{T}$.