Computing $e^{tA}$, where $A$ is a matrix 
Given $$A = \begin{bmatrix} 0&-1\\1&0\end{bmatrix}$$ find $e^{tA}$, where $t \in \mathbb{R}$.

I have read about calculating the matrix exponential here. I know that when $A$ is written in its diagonal form it's kind of easy. The same with nilpotent matrices. Unfortunately, I don't know how to deal with $t$. Especially when the eigenvalues are complex (in the given example they are).

For this special case, one can start out from the observation that $A$ acts on the standard basis of $\Bbb R^2$ exactly as does multiplying by $i$ on the basis $1,i$ of $\Bbb C$. 
In particular, $A^2= -\mathrm{Id}, \ A^3=-A$ and $A^4=\mathrm{Id}$. 
Thus, calculating $e^{tA}$ is formally exactly the same as calculating $e^{ti}$, only that we write $\mathrm{Id}$ in place of $1$ and $A$ in place of $i$, yielding 
$$e^{tA} =\cos t\cdot\mathrm{Id} \ +\sin t\cdot A$$
