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
\[\begin{equation} \begin{aligned} \frac{dN_2}{dt} &= - \frac{dN_1}{dt} - \frac{dN_3}{dt} \\ &= \lambda_1 N_1 - \lambda_2 N_2 \end{aligned} \end{equation}\]

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
\[ \begin{pmatrix}dN_1\\dN_2\end{pmatrix} = \left( \begin{array}{cc} - \lambda_1 & 0 \\ \lambda_1 & - \lambda_2 \end{array} \right) \begin{pmatrix}N_1\\N_2\end{pmatrix} \]

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