Liouville's Theorem (Differential Equations)

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


 * $$\det \Phi (t) = e^{ \int_{t_0}^t \mathrm{tr} A(s) ds } \det \Phi (t_0 ) $$.