What is the order of the multiplicative group $$G = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a,b,c,d \in \mathbb{Z}/31\mathbb{Z} , \, ad - bc \neq 0 \text{ in } \mathbb{Z}/31\mathbb{Z}\right\} \, \text{?}$$
I thought it was natural to phrase the problem in terms of congruences: count the number of lists $(a, b, c, d)$ of integers in $ \{0, 1, \dots, 30\}$ such that $ad \not \equiv bc \, \pmod{31}$. But I'm having trouble going further from here.
When I was a student, I once was assigned the problem of computing the order of ${\rm GL}_2(\mathbf F_p)$ for a particular prime $p$ and did what you tried to do, which for me was successful because the prime was smaller than $31$. But it was still tedious! And ultimately this is the wrong way to think about the problem.
Here is a better approach. When $F$ is a field, a $2 \times 2$ matrix in ${\rm M}_2(F)$ is invertible exactly when its columns are a basis of $F^2$. So what you want to do is count the number of pairs of linearly independent vectors in $\mathbf F_{31}^2$: how many nonzero vectors can be the first term in a basis, and once you pick a nonzero vector $v_1$ in $\mathbf F_{31}^2$, how many vectors $v_2$ are linearly independent of $v_1$ and thus can be the second member of a basis whose first member is $v_1$?
I suggest thinking about this task not just for the specific case of $\mathbf F_{31}^2$, but in $\mathbf F_p^2$ where $p$ is a general prime number. The count of bases of $\mathbf F_p^2$ will depend on $p$.
