Let $W$ be the Wronski matrix of a fundamental system of the homogeneous linear differential equation system $$ y^{(k)}+A_{k-1}(x) y^{(k-1)}+\ldots+A_1(x) y^{\prime}+A_0(x) y=0, $$ where the $A_j$ are continuous functions on an interval with values in the space of $n \times n$-matrices. Then $$ (\det W)^{\prime}+\operatorname{tr} A_{k-1} \det W=0 . $$
I was able to solve this problem. I leave it open for curious readers. A hint is the comment below this question.