## A Two Stage Radioactive Decay Series in Matrix Form

We can represent a radioactive decay series consisting initial of a fixed number of atoms
$N$
, a number of initial stage nuclides
$N_1$
, decaying via an intermediate nuclide
$N_2$
to a stable nuclide
$N_3$

Th initial stage nuclei decay according to the equation
$\frac{dN_1}{dt} =\lambda N_1$

Th intermediate stage nuclei decay according to the equation
\begin{aligned} \frac{dN_2}{dt} &= - \frac{dN_1}{dt} - \frac{dN_3}{dt} \\ &= \lambda_1 N_1 - \lambda_2 N_2 \end{aligned}

Th end stage nuclei are created according to the equation
$\frac{dN_3}{dt} =\lambda_2 N_2$

We can write this in matrix form as
$\begin{pmatrix}dN_1\\dN_2\\dN_3\end{pmatrix} = \left( \begin{array}{ccc} - \lambda_1 & 0 & 0 \\ \lambda_1 & - \lambda_2 & 0 \\ 0 & \lambda_2 & 0 \end{array} \right) \begin{pmatrix}N_1\\N_2\\N_3\end{pmatrix}$

In fact we can reduce the above matrix using the relationship
$N=N_1+N_2+N_3$
. We obtain
$\begin{pmatrix}dN_1\\dN_2\end{pmatrix} = \left( \begin{array}{cc} - \lambda_1 & 0 \\ \lambda_1 & - \lambda_2 \end{array} \right) \begin{pmatrix}N_1\\N_2\end{pmatrix}$