Liouville's Theorem (Differential Equations)

Theorem
If $\Phi \left({t}\right)$ is a solution to the matrix differential equation $X' = A \left({t}\right)X$, with $A \left({t}\right)$ continuous on the interval $I$, and $t_0 \in I$, then:


 * $\det \Phi \left({t}\right) = e^{\int_{t_0}^t \mathrm{tr} A \left({s}\right) \mathrm d s} \det \Phi \left({t_0}\right)$