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!

We could use the characteristic polynomial, if you want.  In this case, we calculate
$$
p(x) = x^2 + 1
$$
By the Cayley-Hamilton theorem, we have
$$
A^2 + I = 0 \implies\\ A^2 = -I
$$
It follows that $A^3 = -A$ and $A^4 = I$.
In general, it would be more useful to find the minimal polynomial of such a matrix.  In this case, the two coincide.
