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What do we know about invertible matrices over $\mathbb{Z}$ or number fields $K$ (or even $\mathbb{Q}$) with inverses in the same ring/field? In the case where $K = \mathbb{Q}$, it is known that matrices with entries in $\mathbb{Q}$ which are invertible have inverses which also have entries in $\mathbb{Q}$. It is also clear that they contain the set of unimodular matrices, which are the matrices with integer entries with determinant $\pm 1$. As for non-diagonal invertible matrices with entries in $K$, I really don't know what they look like.

Is there anything known about estimating how many invertible matrices there are with entries of bounded size in $\mathbb{Z}$ or $\mathbb{Q}$ (assuming the fractions are in lowest terms)? For a lot of objects used in number theory, there seem to be a lot of computational work done related them. However, I couldn't find any references related to counting these objects (or even unimodular matrices).

Any suggestions for how to count these kinds of objects (in the case of $\mathbb{Q}$ or $\mathbb{Z}$) or how to characterize these kinds of matrices (e.g. some idea of entries or group structure)? Any reference related to this (or even estimates) would be helpful.

Edit: Added clarification on counting matrices. The aim of the question is to get an idea of what these matrices look like and how frequently they occur in some sense. Also, the definition of the ``size'' of an element $\frac{p}{q}$ of $\mathbb{Q}$ used here is $\max(|p|, |q|)$ (think of heights of rational points in projective space).

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    $\begingroup$ What do you mean by counting in this context? There are clearly infinitely many such (diagonal matrices alone are infinite). $\endgroup$
    – lulu
    Jul 31, 2017 at 10:54
  • $\begingroup$ Thanks for the comment! I meant counting the number of matrices with entries below a certain size/norm. I'll add this to the question for clarification. $\endgroup$
    – modnar
    Jul 31, 2017 at 10:57
  • $\begingroup$ Even so...as the determinant is a polynomial in the entries, every open neighborhood of a point in $\mathbb Q^n$ will contain the matrix entries of an invertible matrix. Is it possible you meant to count matrices with integer entries with bounded size? $\endgroup$
    – lulu
    Jul 31, 2017 at 11:01
  • $\begingroup$ Yes. I was also wondering if there was some way to characterize invertible matrices over a number field, but it looks like these two things got mixed up in the question. $\endgroup$
    – modnar
    Jul 31, 2017 at 11:03
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    $\begingroup$ I would look at the HNF $\endgroup$
    – reuns
    Jul 31, 2017 at 11:12

1 Answer 1

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Have you heard of the adjugate matrix?$\DeclareMathOperator{\adj}{adj}$

Let $R$ be a commutative ring and $A \in R^{n\times n}$ for some $n \geq 1$. The adjugate matrix of $A$ is $\adj(A)=C^T$, where $C$ is the cofactor matrix of $A$. It is well known that $A$ is invertible if and only if $\det(A)$ is invertible in $R$. In this case, we have

$$A^{-1}=\frac{1}{\det(A)}\adj(A)$$

From this expression, it is clear that $A^{-1}$ also has entries in $R$.

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