Markov Chain Basic Probability Question

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Trying to revise for my upcoming exams however our university doesn't give any solutions so it makes revision rather difficult.

I've got this question: https://gyazo.com/835a68414bc7b52ded90501b86b9442b

I've attempted part a but I may have gone about it wrong, Given X_0 = G

We are looking for P_gw * P_wr

My only confusion is whether I need to work out the matrix P^2?

I have done that and therefore my answer would be 0.34 * 0.17 = 0.0578

Any guidance on part b would be extremely helpful as well!

I apologise that my matrix isn't formatted but I am unsure how to input my 4x4 matrx into LaTeX, if anyone would be able to do it for me that would be greatly appreciated!