Hello,

I have a practice exam paper and I have 3 questions which i have no idea how to do or what the answers are. I somebody could explain them that would be great, I'm desperate!

Q1. Consider the diagram G with three vertices P1, P2 and P3 and four directed edges, one each from P1 to P2, P1 to P3, P2 to P3 and P3 to P1.

a) sketch the diagraph (i did that one)

b) find the adjacency matrix M of G

c) Use the adjoint formula for inverse to find g(z) = (1-zM)^-1

d) expand the (1,3) entry of g(z) as a power series up to terms of order 6.

e) find the number of walks of length 6 from P1 to P3.

Q2. Let W= spam(S) where S = {A1, A2, A3} with A1 = [1 0/0 0], A2 = [0 0/0 1] and A3 = [0 1/1 0].

a) Indetify W in more farmiliar terms (???)

b) Prove that S is a basis of W

c) Find the dimension of W.

d) Find one or more additional matrices A4, A5,...such that T = {A1, A2, A3, A4, A5...} is a basis of M22. Justify your answer.

e) For the matrix X = [2 3/3 4], find the coordinate vectors [X]s and [X]t.

I know its a lot but i have had a few equations i was stuck and am slowly working my way through but i don't know where to begin on these ones. Apparently these will be the hardest ones on the exam and worth the most.

Any help will be vastly appreciated! Thank you

Tania