Finding the order of an element in $GL(2,\mathbb F_p)$. Let $\mathbb F_p=\mathbb{Z}/p\mathbb{Z}$. I am looking for a general method of finding the order of an element in $GL(2,\mathbb F_p)$. Suppose $p=7$ and I am given the element $\begin{pmatrix} 5 &1 \\ 1& 1\\\end{pmatrix}$, and I am asked to find the order, then how should I proceed. One way to proceed is to calculate $A^k$ for each $k$ and see when it takes the value $I_2$ for the first time. But this becomes a tedious task when the order is large. I think there is some other way to figure out the order by some group representation technique, but I do not know that topic very well. Can someone help me?

The eigenvalues are $\lambda_{1,2}=3\pm\sqrt5$, so your matrix is similar to
$$
D=\left(\begin{array}{cc}3+\sqrt5&0\\ 0&3-\sqrt5\end{array}\right).$$
As $5$ is not a quadratic residue modulo $7$ these reside in the extension field $\Bbb{F}_{49}$.
Because $\lambda^2=3^2\pm6\sqrt5+5=\mp\sqrt5$, it follows that $\lambda_{1,2}^4=5$. As $5$ is a primitive (=sixth) root modulo $7$, these calculations imply that the eigenvalues, and hence also he matrix $D$, have order $24$. Similar matrices share the same order, so we are done.
