Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5

Question

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5. I will just add that this task is slightly ahead of my knowledge of field theory. So any pointers would be appreciated.

Answer 1

The top row has to be a nonzero vector.

The 2nd row has to not be a scalar multiple of the top row.

The bottom row has to not be a linear combination of the other two rows.

You should be able to count these things, and combine them appropriately.

Answer 2

Hint:

Take a vector space $\;V\;$ over $\;\Bbb F_5\;$ s.t. $\;\dim V=3\;$. How many basis (ordered ones, of course) does $\;V\;$ have?

For exaple, to make calculations simpler you could take the elementary group $\;\Bbb F_5\times\Bbb F_5\times\Bbb F_5\;$ with the obvious scalar product...