\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
We consider a Markov chain \((X_n)\) with three states: \(A, B,\) and \(C\). The transition probabilities are modeled by the probabilistic graph below. The system starts in state \(A\), so \(\pi_0 = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix}\).
  1. Determine the transition matrix \(\mathbf{M}\) associated with \((X_n)\) using the order \((A, B, C)\).
  2. What is the particularity of state \(C\)?
  3. Using a calculator, calculate the probability vectors \(\pi_5\) and \(\pi_{10}\) (round to 3 decimal places).
  4. What can you conjecture regarding the steady state distribution of this system?

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