the rate equations can be written in matrix form:

$\frac{d N}{dt} = R \cdot N$

the most direct way to solve it is:

$N(t)= Exp(t R) \cdot N(0)$

from the last post. we knew:

$N(t)=E \cdot Exp(t D ) \cdot E^{-1} N(0)$