# How many 2x2 matrices with module 27 inputs exist with the condition being invertible?

I've been checking the following discrete mathematics exercise:

How many 2x2 matrices with module 27 inputs exist with the condition being invertible?

Checking many information in the internet I found that one answer proposed is $$526178$$, another one is $$314928$$ but through different calculations I've been unable to find those numbers.

Any help will be really appreciated

• For a similar question see here. – Dietrich Burde Oct 13 '18 at 18:41

Let the matrix have rows $$a, b$$ and $$c, d$$.

If the matrix is invertible mod $$27$$, then no row or column can be a vector multiple of $$3$$. There are two cases:

Case 1: If $$3\mid a$$ then neither $$b$$ or $$c$$ can be a multiple of $$3.$$ There are $$9$$ choices for $$a$$ and $$18$$ for each of $$b$$ and $$c$$. For a given choice of $$a, b,$$ and $$c$$, we need to exclude any values for $$d$$ that makes $$ad-bc$$ divisible by $$3$$. But $$bc$$ is not divisible by $$3$$ while $$a$$ is. So the congruence $$ad\equiv bc \pmod{3}$$ has no solutions. Therefore $$d$$ can be any of the $$27$$ numbers. We have $$9\cdot 18\cdot 18\cdot 27 = 78732$$ possibilities for Case 1.

Case 2: If $$3\nmid a$$, then $$b$$ and $$c$$ can be anything. So we have $$18$$ choices for $$a$$ and $$27$$ each for $$b$$ and $$c$$. Since $$3\nmid a$$, the congruence $$ax=cd \pmod{3}$$ has a unique solution which lifts to $$9$$ solutions $$\pmod{27}.$$ These must be excluded, so there are $$18$$ choices for $$d$$. We have $$18\cdot 27 \cdot 27\cdot18 = 236196$$ possibilities for Case 2.

We add the two cases and get $$314928$$ for the final answer.

Idea: The number of all $$2\times 2$$ matrices is $$27^4$$. Calculate the number of matrices that are not invertibile, so $$27\mid ad-bc$$ if $$a,b,c,d\in \mathbb{Z}_{27}$$ and

$$M= \left[ \begin{matrix} a&b\\ c&d\\ \end{matrix} \right]$$

Divide it on cases

If $$3\nmid d$$ then $$a$$ is uniqely determined with $$b,c,d$$ with formula $$a= bcd^{-1}{\mod 27}$$

If $$3\mid d$$ and $$9\nmid d$$ ...

If $$9\mid d$$ and $$27\nmid d$$ ...

If $$27\mid d$$ ...