A Markov chain has 3 possible states: A, B, and C. Every hour, it makes a transition to a different state. From state A, transitions to states B and C are equally likely. From state B, transitions to states A and C are equally likely. From state C, it always makes a transition to state A. (a) Write down the transition probability matrix. (b) If the initial distribution for states A, B, and C is P0 = ( 1 3 , 1 3 , 1 3 ), find the distribution of state after 2 transitions, i. e., the distribution of X2. (c) Show that this is a regular Markov Chain. (d) Find the steady-state distribution for this chain.