Could someone please help me to understand what questions a) and b) here are asking for?
Pretty much means what it says:
It wants you to prove that (1) is a solution to the DE given, and then prove that (1) is actually the general solution.
Okay well for a) I'm guessing you would do this by deriving the eigen-equation.
But for b) how would you show that it's not just 'a' solution but is the general solution?
There is no need to derive anything - the equations are given to you. Though you do need to be able to express the relationship between A and it's eigenvectors algebraically.
Use the definition of "general solution".
What is the difference between a specific solution and the general solution of any DE?
All this is stuff you should have covered way back when you 1st learned about 1st and 2nd order ODE's - before you ad to deal with systems of ODEs like above. It is just the same.
Separate names with a comma.