Radiactive Decay With Alternative Decay Routes in Matrix Form

We can represent a radioactive decay series via two possible routes. There is initially a number of atoms
$N$
of nuclide 1. Suppose the two possible decay routes are~
$1 \rightarrow 2 \rightarrow 4$

or
$1 \rightarrow 3 \rightarrow 4$

Th initial stage nuclei 1 decay according to the equation
$\frac{dN_1}{dt} =-\lambda_{12} N_1 -\lambda_{13} N_1$

Where the subscript
$N_{ik}$
denotes decay from state i to state k.
Th intermediate stage nuclei 2 are created according to the equation
$\frac{dN_2}{dt} = \lambda_{12} N_1 -\lambda_{24} N_2$

Th intermediate stage nuclei 3 are created according to the equation
$\frac{dN_3}{dt} = -\lambda_{13} N_1 - \lambda_{34} N_2$

The end stage nuclei 4 are created according to the equation
$\frac{dN_4}{dt} = \lambda_{24} N_2 + \lambda_{34} N_3$

We can write this in matrix form as
$\begin{pmatrix}dN_1\\dN_2\\dN_3\\dN_4 \end{pmatrix} = \left( \begin{array}{cccc} - \lambda_{12} - \lambda_{13} & 0 & 0 & 0 \\ \lambda_{12} & - \lambda_{24} & 0 & 0 \\ 0 & \lambda_{13} & - \lambda_{34} & 0 \\ 0 & \lambda_{24} & \lambda_{34} & 0 \end{array} \right) \begin{pmatrix}N_1\\N_2\\N_3\\N_4\end{pmatrix}$

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