If $\Phi(x)$ is the fundamenta solution matrix of system $\dot{x}=Ax$,and $\Phi(0)=E$ ( Identity matix) , what is $\Phi^{-1}(x)$?

Question

If $\Phi(x)$ is the fundamenta solution matrix of system $\dot{x}=Ax$,($A$ is constant matrix) and $\Phi(0)=E$ ( Identity matix) , what is $\Phi^{-1}(x)$ ?

I intend to find expression of $\Phi^{-1}(x)$. By $\Phi(0)=E$, we can get $\Phi(x)\cdot\Phi^{-1}(x_0)=\Phi(x-x_0)$ by uniqueness theorem. But I am confused how to get another step. Any hints would be helpful.

Answer 1

$\Phi(x)=e^{xA}E$, $\Phi^{-1}=E^{-1}e^{-xA}=e^{-xA}$ if $E$ is the identity matrix.