Find A from $e^{At}$

Question

I have been given a fundemanetal matrix $e^{At}$ and am asked to find $A$. It's kind of an odd question as we've mainly been calculating $e^{At}$ when given a certain matrix $A$. My question is how do I find $A$ and if you can always find $A$ for any fundamental matrix. Anyways this was given:

\begin{align} e^{At}&= \begin{bmatrix} 2e^{2t}-e^t & e^{2t}-e^t&e^t-e^{2t} \\ e^{2t}-e^t & 2e^{2t}-e^t&e^t-e^{2t} \\ 3e^{2t}-3e^t & 3e^{2t}-3e^t&3e^t-2^{2t} \\ \end{bmatrix} \end{align}

Answer 1

Hint: Take the derivative towards $t$ and plug in $t=0$.