Liouville's Theorem (Differential Equations)

Theorem
Let $\Phi \left({t}\right)$ be a solution to the matrix differential equation:
 * $X' = A \left({t}\right)X$

with $A \left({t}\right)$ continuous on the interval $I$ such that $t_0 \in I$.

Then:


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