13. (Bonus Problem - max 5 points - show all work) In a certain town, it has been observed that . if a day is sunny, then it has 70% chance of being followed by another sunny day, . if a day is cloudy, then 4 out of 5 times it is followed by another cloudy one. (a) Let S() and C() represent the probability of the ith day being sunny or cloudy, respectively Find the 2 x 2 matrix A such that ( 80+) ] - 4 [50 ]. (b) Show that A is a regular matrix. (c) Show that A is a stochastic matrix. (d) Starting with a sunny day, i.e., | coo ]- [ : ]. calculate the probability vectors S(1) C (1) for i = 1, 2, 3. (e) Starting with a cloudy day, calculate the probability vectors C(i ) for i = 1, 2, 3. f) Find the stable vector of A and compare it to the vectors obtained in parts (d) and (e)