Powers of a Markov matrix

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Why are the powers of a Markov matrix also markov?

Markov matrix has all entries >=0 and all columns sum to 1.

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HINT: $A$ is a Markov matrix if and only if its entries are nonnegative and $A^\top\begin{bmatrix}1\\1\\\vdots\\1\end{bmatrix}$ is what? (And, after that, what is the transpose of $A^k$?)