Tom and Jerry move independently back and forth between two rooms. At each time step, Tom moves from the current room to the other room with probability 0.8. Starting from room 1, Jerry moves to room 2 with probability 0.3 and otherwise remains in the room. Starting from room 2, Jerry moves to room 1 with probability 0.6 and otherwise remains in the room.

(a) Tom's location (i.e., the number of the room where Tom is) can be described by a Markov chain. Find the transition matrix of this chain.

(b) Jerry's location (i.e., the number of the room where Jerry is) can be described by a Markov chain. Find the transition matrix of this chain.

(c) The locations of both Tom and Jerry at each time step can be described by a Markov chain with 4 possible states: both in room 1, Tom in room 1 and Jerry in room 2, Tom in room 2 and Jerry in room 1, and both in room 2. Label these states as 1, 2, 3, 4, respectively and let Xn be the number representing the (Tom's location, Jerry's location) state at time n. Find the transition matrix of the chain Xn.