Finding the matrix for a matrix exponential

As you know $$e^{tA} = I + tA + (t^2/2)A^2 + ....$$

Thus if you differentiate $$e^{tA}=\begin{pmatrix} \frac{1}{2}(e^t+e^{-t}) & 0 & \frac{1}{2}(e^t-e^{-t}) \\ 0 & e^t & 0 \\ \frac{1}{2}(e^t-e^{-t}) & 0 & \frac{1}{2}(e^t+e^{-t}) \end{pmatrix}$$ and evaluate the result at $t=0$, you will get your matrix $A$

I found $$A= \begin{pmatrix} 0&0&1\\0&1&0\\1&0&0\end{pmatrix}$$

Notice that the matrix $A$ satisfies $$(e^{tA})' = Ae^{tA}$$ and $$ e^{tA} =I $$ at $t=0.$