Let $A(t)$ and $B(t)$ be matrices with each element in $L^\infty(0,T).$ Let $A(t)$ have an inverse. I know nothing else about this inverse.

Let $c(t)$ be a vector in $L^2(0,T).$

Let $u(t) \in \mathbb{R}^n$ be unknown. Does the system $$A(t)u'(t) + B(t)u + c(t)= 0$$ have a unique solution?

Again, I don't know that $A(t)^{-1}$ is in $L^\infty.$