Counting Invertible Matrices in GL(3, Z_2)

[Treadstone 71]

Is there a systematic way of counting the number of invertible matrices in a general linear group with entries in a finite ring? For example, GL(3, Z_2). The determinant has to be zero, but other than that, I don't know any systematic way of counting them. I usually start by saying that there are at least 13 non-invertible ones (if at least one row or column are zeros) then I look at the equation of the determinant and try to go from there.

 

[matt grime]

As long as you mean 'over a finite field' the answer is yes, since invertible is the same as linear independence of rows (or columns)