Here's what I would have submitted:

Question 1:

L^n * v0 = 1/(2sqrt5)L^n(v2 -v1) = 1/(2sqrt5)(L^n v2 - L^n v1)
=1/(2sqrt5)(L^(n-1) L v2 - L^(n-1) L v1)
=1/(2sqrt5)(L^(n-1) t2 v2 - L^(n-1) t1 v1)
=1/(2sqrt5)(L^(n-2) t2^2 v2 - L^(n-2) t1^2 v1)
=...
=1/(2sqrt5)(t2^n v2 - t1^n v1). 



Question 2:

This is 
[0 1 0 0 0]
[0 0 2 0 0]
[0 0 0 3 0]
[0 0 0 0 4].



Question 3:

We know that one eigenvalue of O is 2+i, with corresponding
eigenvector the first column of P.  Applying Theorem 9, we get 
B=[-1 1] and C=[ 2 1]
  [ 1 0]       [-1 2]