Consider a continuous and locally integrable matrix $M:t\mapsto M(t) \in \mathcal{M}_{n,n}(\mathbb{R} \mbox{ or } \mathbb{C})$
I want to find a $\mathcal{C}^1$ function $A:t\mapsto A(t) \in GL_n^+(\mathbb{R}) \mbox{ or } GL_n(\mathbb{C})$ satisfying
$$ A'(t) \times adj (A(t)) = M(t)$$
where $ A'$ is the usual derivative of $A$ and $adj(A(t))$ is the adjugate matrix of $A(t)$, that is to say the transpose of the cofactor matrix.
Of course such a function might not always exist.
Has this problem already been studied ? If so, what kind of reference should I consult ?
If such a problem has not been studied yet, hopefully we can still come up with a solution for $n=2$