Part (a) With the states numbered as in the diagram, the transition matrix is

P = 0 1/2 1/2 0 1/2 0 0 1/2 1/2 0 0 1/2 0 1/2 1/2 0 .

Using the methods in the text, we calculate the steady state vector as

v = [1/4   1/4   1/4   1/4].

Part (b) If you calculate P², you find that P² = P. Thus, all powers of P are equal to P. Since there are zero entries in P, it follows that there will be zero entries in all powers of P. Hence, P is not regular.

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