Find the order of this matrix on the group $(GL_{2}(\mathbb{C}),\cdot)$. I have to calculate the order of the matrix 
\begin{equation}
A=
  \left( {\begin{array}{cc}
   i & 0\\
   -2i & -i\\
  \end{array} } \right)
\end{equation}
on $(GL_{2}(\mathbb{C}),\cdot)$.
I found, by calculating it doing $A^4$, that the order is $4$, but my question is:
Is there any way to prove it more formally, using its characteristic polynomial, or by another way?
Thank you!

The first thing that comes to my mind is to diagonalize the matrix and note that 
$$A^n = Q^{-1} \Delta^n Q $$
where  $\Delta$ is the diagonal matrix contaning the eigenvalues and $Q$ the ortoghonal matrix realizing the change of coordinate. As you have $A^n = I\iff \Delta ^n = I$ you can simply calculate eigenvalues of $A$ and check for the order of $\Delta$, that is much less time consuming.
