(3) (24 points) A certain college dining hall tends to serve pasta a lot as the main dish. Other main dish items are turkey with gravy and beef au jus, but only one main dish is served each night. After eating at this dining hall frequently, you determine the following facts: If pasta is served on a given night, there is a 75% chance that it will be served the next night. If turkey is served on a given night, there is a 25% chance that it will be served the next night. Beef au jus is never served two nights in a row. If pasta is served on one night, there is an equal chance that turkey with gravy or beef au jus will be served the following night. If turkey with gravy is served on one night, there is twice the chance that beef au jus will be served as the chance that pasta will be served on the following night. If beef au jus is served on one night, it is four times as likely that pasta will be served as that turkey with gravy will be served on the following night. (Please use technology for this problem.) (a) Write out the transition matrix for the associated Markov chain. sition matrix regulur? Why or why not? (c) Suppose that pasta is served on Monday. What is the probability that it is also served on Thursday? (d) What happens in the long term? Does this Markov chain have a steady-state vector? If so, find it. If not, explain what happens instead.