Consumers in Shelbyville have a choice of one of two fast food restaurants, Krusty's and McDonald's. Both have trouble keeping customers. Of those who last went to Krusty's, 57% will go to McDonald's next time, and of those who last went to McDonald's, 81% will go to Krusty's next time.

( a) Find the transition matrix describing this situation. (Assume the components of the state vector are in this order [Krusty's customers, McDonald's customers]).
$\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$

( b) A customer goes out for fast food every Sunday, and just went to Krusty's.

i. What is the probability that two Sundays from now she will go to McDonald's?

ii. What is the probability that three Sundays from now she will go to McDonald's?

( c) Suppose a consumer has just moved to Shelbyville, and there is a 32% chance that he will go to Krusty's for his first fast food outing. What is the probability that his third fast food experience will be at Krusty's?

( d) Find the steady-state vector.
$\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$