First, Let me define a finite field $\mathbb Z_p $ (p is prime) with the following addition and multiplication:
$\ a,b \in \Bbb Z \\$ \ $\ a + b = $ remainder of $\ a +b $ when diving them by $\ p$
$\ a \times b = $ remainder of multiplication when dividing by $\ p $
Basically same as GF(2) field but with different number.
Let $\ \mathbf M_{2 \times 2} (\mathbb Z_p) $ group of matrices and I need to find the number of invertible matrices in that group.
I was thinking maybe to try and find under what conditions the determinant is zero but I couldn't figure it out on my own.
Best Answer
Hint:
the first row has to be a non-zero vector in $\mathbf F_p^2$, which makes $p^2-1$ possibilities.
This vector being chosen, the second row is another vector which must be non-collinear to the first, which eliminates its $p$ scalar multiples (these include the zero vector).
Note:
Following the same lines, you can show the number of invertible $n\times n$-matrices in $\mathbf F_p$ is $$ \bigl|\operatorname{GL}(n,\mathbf F_p)\bigr|=\prod_{i=0}^{n-1}(p^n-p^{i}).$$