Correct answer: (A) π is stationary.; (B) πᵀ is an eigenvector of Pᵀ.; (C) P(X₃=1|X₁=1)=11/30
Explanation
Multiplication gives πP=(24/3+25·2/5,18/2+25·3/5,18/2+24·2/3)/67=(18,24,25)/67, proving A and the eigenvalue-1 statement B. The two-step return probability is (1/2)(1/3)+(1/2)(2/5)=1/6+1/5=11/30. All states communicate in a finite chain and are positive recurrent, refuting D. **A,B,C**.
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