How to prove that the column sum for a markov matrix is 1?

Question

As is the topic, it is obvious and easy to explain in non-math language but how do I mathematically prove it?

Answer 1

Assume we have a state space $S = \{1,2, \ldots, k\}$. Then the sum of the elements lying on $m-$ row of the transition matrix is:

$$\sum_{i=1}^k p_{mi}\begin{array}[t]{l}=p_{m1}+p_{m2}+\cdots+p_{mk} \\ =prob\{X_n=1\, \mid X_{n-1}=m\}+\cdots+prob\{X_n=k\, \mid X_{n-1}=m\} \\ =prob\left\{(X_n=1)\cup (X_n=2) \cup \cdots \cup (X_n=k)\, \mid X_{n-1}=m\right\} \\ =prob\{X_n \in S\, \mid X_{n-1}=m\} \\ =1. \end{array}$$

P.S. I assumed that the transition matrix is a row - stochastic matrix.