Descrete time Markov chain

Question 1. Find the solution to the following system of equations r + 2y + 2 = 1 3r + 7y + 32 =1 -2c - 31 -42 =1 using (a) (2 points) Naive Gaussian elimination, (b) (3 points) Gaussian elimination with partial pivoting, (c) (4 points) Gaussian elimination with complete pivoting, and (d) (4 points) Gaussian elimination with scaled partial pivoting. Make sure to clearly show all your steps, including any row/column exchanges and index/scaling vectors.Consider a Markov chain {Xn, n = 0, 1, . .. } on the state space S = {0, 1, 2}. Suppose that the Markov chain has the transition matrix 0 P = 1 0 0 NIH 1. Show that the state space is irreducible. 2. Show that the Markov chain is periodic. Find the period of the Markov chain. 3. Let h denote a stationary mass of the Markov chain. Find h(x) for all x E S.4. Consider a discrete-time Markov chain with the following probability transition matrix 0 0 P= I-T- VVO 0 0 1 Is it possible to choose values for ar and y so that the Markov chain has the following properties? In each case, state the values of a and y, or give a brief reason why it is not possible. (a) The Markov chain has period 2. (b) The Markov chain is reducible. UNNN (c) The Markov chain has at least one transient state. (d) The Markov chain has invariant distribution (1/4, 1/4, 1/4, 1/4)