It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$ It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way that leaps to mind of constructing an element of order $p+1$ is by constructing the companion matrix of the irreducible polynomial whose root is the multiplicative generator of $GF( p^2)$ and then raising it to $p-1$st power - this is quite non-explicit. I assume there is no explicit construction which works for all $p,$ but I could be wrong: is there?