This problem has been giving me a lot of trouble. Let $\mathbb{F}_q$ be the finite field of order $q$. Let $A\in SL(2,\mathbb{F}_q)$, the multiplicative group of $2\times 2$ matrices with determinant $1$. Note that the characteristic equation of $A$ is $x^2-\text{tr}(A)x+1=0$. Show that if the characteristic equation for $A$ has distinct solutions in $\mathbb{F}_q$, then $|A|$ divides $q-1$, and if the characteristic equation of $A$ does not have any solutions in $\mathbb{F}_q$, then $|A|$ divides $q+1$.
A piece of useful information is the fact that $SL(2,\mathbb{F}_q)$ has an order of $q^3-q$. Showing that $|A|$ divides $q\pm 1$ is equivalent to showing that $A^{q\pm 1}=I_2$, the $2\times 2$ identity matrix. I'm not sure where to go from here though. I get the impression this problem may involve field extensions as well.
If the (distinct) eigenvalues $\alpha$ and $\beta$ lie in $\Bbb F_q$ then $\alpha^{q-1}=\beta^{q-1}=1$.
If $\alpha$ and $\beta$ are outside $\Bbb F_q$ then $\beta=\alpha^q$ (Frobenius automorphism) and $\alpha\beta=1$ (determinant).