Liouville's formula

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 $Φ (t)$ . Under standard assumptions we have

det Φ (t) = det Φ (t_{0}) \cdot e^{\int_{t_{0}}^{t} tr A (s) d s} .