Find the order of this matrix on the group $(GL_{2}(\mathbb{C}),\cdot)$.

Question

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!

Answer 1

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.

Answer 2

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.