linear algebra - Integer matrices with determinant equal to $1$

Integer matrices with determinant equal to $1$ are quite useful in many situations. Take, for example, this question. For the $2 \times 2$ case it's easy to find many such matrices, e.g.,

$$\begin{bmatrix}
2 & 3 \\
3 & 5 \\
\end{bmatrix}$$

$$\begin{bmatrix}
4 & 3 \\
5 & 4 \\
\end{bmatrix}$$

• But how to construct the procedure for generation integer matrix with
  arbitrarily chosen dimension $n \times n$?


• Is it a method which is as general as it is possible?


• I'm also interested in the answer how many degrees of freedom has an
  
  integer matrix with determinant equal 1 (or other perhaps number) ?
  Without determinant constraint $n \times n$ matrix has of course $n^2$ degrees of freedom.. how many is lost when we constrain it with determinant?

Answer

You can just start with the identity matrix and apply transformations that don't change the determinant:

1. Adding to column (row) another column (row) multiplied by an integer.


2. Performing an even permutation of the columns (rows).

Hart to tell what is degrees of freedom for a discrete set. Its dimension is zero.
But you can think of it a cutting all the $n^2$ dimensions that you had by one equation. So, $n^2-1$.