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$S_1$ is the state vector after one iteration of the Markov chain. Its top entry is the probability of being in state A, and its bottom entry is the probability of being in state B after one step.

Conditions: Two-state chain with states A and B.; Entries are probabilities.; In the examples shown, the entries sum to 1.

$S_1$ is the state vector after one iteration of the Markov chain. Its top entry is the probability of being in state A, and its bottom entry is the probability of being in state B after one step.

Conditions: Two-state chain with states A and B.; Entries are probabilities.; In the examples shown, the entries sum to 1.

No, $S_0 = (1,0)^T$ is not the only possible initial state. It is an asserted initial condition for this specific example.

Conditions: The system is a Markov chain with states A and B.; The entries of the initial state vector are probabilities.; The sum of the probabilities in the initial state vector must equal 1.

No, $S_0 = (1,0)^T$ is not the only possible initial state. It is an asserted initial condition for this specific example.

Conditions: The system is a Markov chain with states A and B.; The entries of the initial state vector are probabilities.; The sum of the probabilities in the initial state vector must equal 1.