All probabilities in a row must sum to 1, as they represent all possible outcomes from that state.
For example, if we have three states (A, B, C), a transition matrix P would look like:
$$P = \begin{pmatrix} p_{11} & p_{12} & p_{13} \\ p_{21} & p_{22} & p_{23} \\ p_{31} & p_{32} & p_{33} \end{pmatrix}$$
Consider the weather forecast:
The transition matrix would be:
$$P = \begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix}$$
The key characteristic of a Markov chain is that it is memoryless. Only the current state matters for predicting the next state.
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