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We add the equation $\pi(a)+\pi(b)=1$ because $\pi$ is a valid probability distribution, meaning its elements must sum to 1. This provides the second independent equation needed to solve the underdetermined system created by the redundant balance equations.

Conditions: $\pi$ is a probability distribution over the states.; The state space has exactly two states, a and b.

We add the equation $\pi(a)+\pi(b)=1$ because $\pi$ is a valid probability distribution, meaning its elements must sum to 1. This provides the second independent equation needed to solve the underdetermined system created by the redundant balance equations.

Conditions: $\pi$ is a probability distribution over the states.; The state space has exactly two states, a and b.

The equations derived from $\pi P = \pi$ are linearly dependent because the rows of the transition matrix $P$ sum to 1. This implies that the sum of the equations is identically zero, leaving only $N-1$ independent constraints for $N$ states.

Conditions: $P$ is a stochastic matrix (rows sum to 1).; $\pi$ is a row vector of probabilities.

The equations derived from $\pi P = \pi$ are linearly dependent because the rows of the transition matrix $P$ sum to 1. This implies that the sum of the equations is identically zero, leaving only $N-1$ independent constraints for $N$ states.

Conditions: $P$ is a stochastic matrix (rows sum to 1).; $\pi$ is a row vector of probabilities.