Q:

How to find A given e^{tA}?

Given I have:
$e^{tA} = e^t \begin{bmatrix}
                    \cos(t)& -\sin(t) \\
                    \sin(t) & \cos(t)
\end{bmatrix}$
How do I find A?
I believe to find A, I need to take the derivative of $e^{tA}$, but I don't see how that yields the original matrix A.

A:

Use $ d(e^{tA}) / dt = A e^{tA}$. You can verify this by looking at the Taylor expansion for $e^{tA}$ (which is legitimate even for matrices!) or, even better, if $A$ is diagonalisable, go to an eigenvector basis where $A$ is diagonal.
Then $ A = d(e^{tA}) / dt |_{t = 0}$.

