A rat runs through the maze shown below. At each step it leaves or stay by choosing at random one of the doors with equal probability. For example, if it is in room 0, it can has two options: stay at room 0 , or go to room 2 with equal probability of 21 for each option. If it is in room 3 , it has three options: stay at room 3 , go to room 2 or room 5 with equal probability of 31 for each option. a) Give the transition matrix P for this Markov chain. b) Find the stationary distribution c) Find the expected time to return to room 0 . d) Now suppose that a piece of mature cheddar is placed on a deadly trap in Room 4. The mouse starts in Room 0. Find the expected number of steps before reaching Room 4 for the first time, starting in Room 0