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:

$\displaystyle x_A(k)=\textrm{proportion of A after k centuries}$

$\displaystyle x_B(k)=\textrm{proportion of B after k centuries}$

$\displaystyle x_C(k)=\textrm{proportion of C after k centuries}$

$\displaystyle x_A(k+1)=0.98x_A(k)$

$\displaystyle x_B(k+1)=0.02x_A(k)+0.99x_B(k)$

$\displaystyle x_C(k+1)=0.01x_B(k)+x_C(k)$

$\displaystyle \mathbf{x}(k+1)=\begin{pmatrix}0.98&0&0\\0.02&0.99 &0\\0&0.01&1\end{pmatrix}\mathbf{x}(k)$

Let $\displaystyle A=\begin{pmatrix}0.98&0&0\\0.02&0.99&0\\0&0.01&1\e nd{pmatrix}$

$\displaystyle \rightarrow \begin{pmatrix}0.98&0&0\\0&0.99&0\\0&0&1\end{pmatr ix}$

I'm not too sure if I'm correct, I think the eigenvalues are $\displaystyle 0.98, 0.99, 1$

At 500 years:

$\displaystyle \mathbf{x}(5)=A^5 \mathbf{x}(0)$

Is what I'm doing correct so far?