I think you have set up the equations and matrix correctly. Now find the eigenvalues and eigenvectors as you said, and this will give you the time dependence of the two eigenvectors ( call them a'(t) and b'(t) ), which are linear combinations of a(t) and b(t). Then you can solve for a(t) and b(t) in terms of a'(t) and b'(t), and since you know the time dependence of a'(t) and b'(t), you will have the time dependence of a and b. Does this make sense?
I have solved the eigen/values/vectors and put them in "general form". So i think using these initial conditions for A(0)=75 and B(0)= 0 i will have solved for a'(t) and b'(t).. Is this correct?
<<<is terrible at math and needs a lot of repetition to understand what is going on.
Thanks for your help so far though phyzguy, it seems like i am kind of on the right track.