I figured out parts a and b. I wrote stchastic matrix (let us call it P) the way I mentioned above.

Assuming probability vector for Monday = X0, tuesday = X1 and so on, I solved part b using

X1 = P X0

X2 = P X1

X3 = P X2

But do not know how to solve part C. I first thought that I have to find a vector X such that

P X = X

But, here we do not use the information that there is an equally likely chance of good, bad, or fair weather today.