a.Construct the 3x3 transition matrix, such that each row gives the probability of each possible outcome, given the current state. 

b.Is the process a Markov chain? In other words, does the conditional probability of the next day's diet depend only on the current state (today's meal) and number of days since the start of time? If not, give an example to demonstrate that the transition probability is path-dependent.

c.Use the rules of matrix multiplication to multiply the transition matrix by itself. What is the probability distribution for the second day?

d.If you continue to raise the transition matrix to higher and higher powers, does the probability distribution converge to a limiting distribution? Show how you reach your conclusion.

Illustrative Example: A university student drinks beer every day, but to ensure a diverse diet, he choose randomly among pizza, burgers and burritos. He eats once per day. His dietary choices conform to these rules:

If I eat burritos today, tomorrow I eat pizza and burgers with equal probability but never burritos

If I eat burgers today, tomorrow I eat burgers w.p. 1/10; pizza w.p. 4/10; and burritos w.p. 5/10

If I eat pizza today, tomorrow I eat burgers w.p. 4/10; burritos w.p. 6/10 and never pizza