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!