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Show that, for every integer $n \geq 2$, the number of $2 \times 2$ matrices with integer entries in $\{0,1,2, \ldots, n-1\}$ and having a determinant of the form $1(\bmod n)$, is $$ n^{3} \cdot \prod_{q \text { prime }, q \mid n}\left(1-\frac{1}{q^{2}}\right) $$. I considered cases of some numbers from lets say $n = 5$ , i observed the pattern that 1mod5 can happen when $ad-bc -1$ is a multiple of $5$ , $(a,d,b,c)$ being $(3,2,0,0)$ seems to satisfy it its looks like there are only some possible combinations which make the ad part greater than bc by just 1 but how to show it ?

Orion_Pax
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  • https://math.stackexchange.com/questions/341033/how-to-calculate-operatorname-sl-2-mathbb-z-n-mathbb-z – Chris Sanders Mar 28 '22 at 07:54
  • @ChrisSanders i am not getting most of the answer given there please can you give a solution which is understandable to a 11-12th grader who knows olympiad maths but not much – Orion_Pax Mar 28 '22 at 08:13
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    Maybe the users of the website "Art of Problem Solving" could provide a solution along the lines of Olympiad Math. The forum is quite popular – Chris Sanders Mar 28 '22 at 08:33

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