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A matrix is introduced to provide an efficient algebraic process for moving from one state vector to later state vectors, such as $S_2$ or $S_100$, rather than re-reading the diagram each time.

Conditions: The Markov chain has fixed one-step transition probabilities.; State vectors are represented in a consistent order.

A matrix is introduced to provide an efficient algebraic process for moving from one state vector to later state vectors, such as $S_2$ or $S_100$, rather than re-reading the diagram each time.

Conditions: The Markov chain has fixed one-step transition probabilities.; State vectors are represented in a consistent order.

Matrices and vectors are introduced as a linear algebra formalism to make manipulations and calculations simpler when studying a Markov chain, especially for iterating states efficiently over many steps.

Conditions: The problem involves a Markov chain.; The goal is to compute future states or manipulate transition probabilities.

Matrices and vectors are introduced as a linear algebra formalism to make manipulations and calculations simpler when studying a Markov chain, especially for iterating states efficiently over many steps.

Conditions: The problem involves a Markov chain.; The goal is to compute future states or manipulate transition probabilities.