Why is the fundamental matrix of a linear system of ODEs always invertible? Why does $\phi^{-1}(0)$ exist, where $\phi(t)$ is the fundamental matrix of the system $\dot{x}=A(t)x$, $x \in \mathbb{R}^n$? I am not able to figure this out.

Consider the ODE $\dot x = - A(T-t) x$.  Its fundamental solution $\psi(t)$ satisfies $\psi(T) \phi(T) = \phi(T) \psi(T) = \text{Identity}$.  (In other words, I am solving the equation $\dot x = A(t) x$ backwards in time.)
