Please answer it correctly and do it hand written . Thank you

10. (Psychology) A behavioral psychologist places a rat each day in a cage with two doors, A and B. The rat can go through door A, where it receives an electric shock. or through door B. where it receives some food. A record is made of the door through which the rat passes. At the start of the experiment, on a Monday, the rat is equally likely to go through door A as through door B. After going through door A and receiving a shock, the proba- bility of going ng through the same door on the next day is 0.3. After going through door B and receiving food, the probability of going through the same door on the next day is 0.6. (a) Write the transition matrix for the Markov process. (b) What is the probability of the rat going through door A on Thursday (the third day after starting the exper- iment)? (c) What is the steady-state vector