Dear forum members,(adsbygoogle = window.adsbygoogle || []).push({});

I have a small problem counting all the invertible matrices of the size 2x2 providing [itex]\mathbb{Z}_{n}[/itex]. This problem was difficult for me so I decided to go on counting how many invertible 2x2 matrices there are for [itex]n=32[/itex]. My strategy to solve the problem was first by calculating all possible matrices and then subtracting all the matrices for which the deteminant is equal to zero.

Given matrix [itex]\begin{pmatrix}a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2}\end{pmatrix}[/itex] in [itex]\mathbb{Z}_{32}[/itex], there are [itex]32^4[/itex] possible matrices.

The cases in which the determinant is equal to zero are:

If [itex]a_{1,1} = 0, a_{1,2} = 0, a_{2,1}=0, a_{2,2}=0[/itex], which is eqaul to one case.

If there are three elements equal to zero and one not, the possibilities of that are [itex]{4 choose 3} \cdot 32[/itex].

If there are two elements equal to zero, provided they arenotdiagonally of each other ([itex](a_{1,1} = 0 \vee a_{2,2} = 0) \wedge (a_{1,2} = 0 \vee a_{2,1} = 0)[/itex]), the number of possibilities of that are [itex]({4 \choose 2} - 2) \cdot 32^2[/itex].

Now how can I continue to calculate the other number of possibilities? I am not sure how can I count the possibilities when [itex]a_{1,1} \cdot a_{2,2} = a_{1,2} \cdot a_{2,1}[/itex].

I have already searched the forum, there was I think one similiar case (see: showthread.php?t=470215), but there [itex]r[/itex] is a prime and I don't actually get how they suddenly got to that formula, furtheremore the comments suggest that the question is ill-formulated.

Furthermore I have made a small program capable of finding the solution of all non-invertible matrices for this question, however strangely it came up with the answer 7600, which is too low I think, but correct me if I'm wrong. It's written in JAVA.

Thank you for your responses!

PS: I cannot post links, so I post a partial link, the begin section should be equal to the link of this forum. If anyone would like to see the program, the pastebin code is 01uKfAKF.

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# Homework Help: Counting invertible matrices of 2x2 provided Zn

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