Let $\phi(t)$ be a fundamental matrix of $\dot x=A(t)x$ on $[a,b]$ and let $C$ be any constant matrix.
- show in general $C\phi$ is not fundamental matrix of $\dot x=A(t)x$ on $[a,b]$.
- detrmine $B$ such that $C\phi$ is a fundamental matrix of $\dot x=B(t)x$ on $[a,b]$.
Since $\phi(t)$ is a fundamental matrix, $\dot{\phi}(t) = A(t) \phi(t)$. Now, consider
$$\frac{d}{dt} \left( C \phi(t) \right) = C \dot{\phi}(t) = C A(t) \phi(t) \neq A(t) (C \phi(t))$$
Hence, $C \phi(t)$ is a fundamental matrix only if $A(t)$ and $C$ commutes, which does not hold in general. For the second question, similarly
$$\frac{d}{dt} \left( C \phi(t) \right) = C \dot{\phi}(t) = C A(t) \phi(t) = B(t) (C \phi(t))$$
Therefore, $B(t)$ is a matrix that satisfies $C A(t) = B(t) C$. Obviously, $B(t) = C A(t) C^{-1}$ if $C$ has an inverse.