A radioactive isotope A decays at the rate of 2% per century into a second radioactive isotope B, which in turn decays at a rate of 1% per century into a stable isotope C.

a) Find a system of linear differential equations to describe the decay process. If we start with pure A, what are the proportions of A, B and C after 500 years, after 1,000 years and after 1,000,000 years?

This is what I've done so far:

Let

I'm not too sure if I'm correct, I think the eigenvalues are

At 500 years:

Is what I'm doing correct so far?