Unimodular matrices of size $n$

Question

Recently, I answered this question on physics stack exchange. So, this motivated me to generated all unimodular matrices of size $n$. For n=1, it is trivial. While for $n=2$, one can pick $\vec{v} = (v_1, v_2)$ such that $gcd(v_1,v_2)=1$ and find $\vec{u} = (u_1,u_2)$ such that $v_1 u_2 - v_2 u_2 = 1$ uniquely for every $\vec{v}$. These vectors are column vectors of unimodular matrix. This generates all unimodular matrices of size 2.

Similar techniques can be used to generate unimodular matrices of size 3. But this brings some questions into my mind about generating all families of unimodular matrices of arbirtary size $n$.

1. I realise that unimodular matrices form a group. So, how can one find the generators of this group ?

Answer 1

The answer to this question has been discovered in the links posted in the comments. I will list my observations and give a sketch of the discovery.

Observations:

1. GCD of elements in each row and each column is 1. So, GCD of all elements of matrix is 1.
2. Applying row and column operations keeps GCD of all elements invariant.
3. Minimum absolute value among non-zero elements can be decreased to 1 by applying row and column operations. (Applying euclid's division lemma appropriately ensures this.)
4. Using observation 2 and 3, any unimodular matrix can transformed into identity matrix.

Since row and column operations are equivalent to multiplying by elementary matrices, elementary matrices generate SL(n,Z).