Also known as Liouville's theorem. See Wikipedia and @chicone2006ordinary (also online.
Given a system of $n$ first-order homogeneous linear differential equation
$$ \dot{y}=A(t)y $$consider a matrix solution $\Phi(t)$. Under standard assumptions we have
$$ \det{\Phi(t)}=\det\Phi(t_0)\cdot e^{\int_{t_0}^t\mbox{tr}A(s) ds}. $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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