In my ODE book (ODE by Wolfgang Walter), I have this exercise.
Let $G(x)$ be a fundamental matrix of $$ y^{\prime}=A(x) y . $$ Show that $\left(G(x)^{-1}\right)^{\top}$ is a fundamental matrix of $$ z^{\prime}=-A(x)^{\top} z. $$
I solved this Problem, but I can still leave it here open as a puzzle.