Eigenvalue of stochastic matrix

Question

Why is the eigenvalue of a stochastic matrix always $1$? I have found lots of articles simply saying it is obvious that the eigenvalue is $1$ but can't get my head around the proofs.

Answer 1

If the rows of $\mathrm P \in \mathbb R^{n \times n}$ sum to $1$, then

$$\mathrm P 1_n = 1_n$$

Hence, $(1,1_n)$ is an eigenpair of $\mathrm P$.