My question is: How do I find the number of $3 \times 3$ matrices $A$ with elements in $F_p$ such that the determinant is non-zero?
I don't really know how to go at it. I have a feeling that maybe Gaussian coefficients does the trick, but I cannot really reformulate the problem in a more understandable way.
This is equivalent to find how many triples of linearly independent vectors there exists in $F_p^3$. There are $p^3-1$ nonzero vectors in $F_p^3$, now for any $v\in F_p^3\setminus\{0\}$ there are $p^3-p$ vectors linearly independent from $v$ and for any pair of linearly independent vectors there are $p^3-p^2$ vectors independent from both of them. So there are $(p^3-1)(p^3-p)(p^3-p^2)$ matrices with determinant nonzero.