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.