Elementary matrices of the first kind generate the special linear group over a field

Statement
Let $$K$$ be a field. Denote by $$SL(n,K)$$ the special linear group over $$K$$: the group of invertible $$n \times n$$ matrices over $$K$$ with determinant equal to $$1$$. Then, $$SL(n,K)$$ is generated by elementary matrices (also called elementary matrices of the first kind or shear matrices), i.e., matrices of the form $$E_{ij}(\lambda)$$, where $$\lambda \in K, 1 \le i,j \le n, i \ne j$$. The matrix $$E_{ij}(\lambda)$$ has $$1$$s on the diagonal, $$\lambda$$ in the $$(ij)^{th}$$ position, and zeros elsewhere.

The proof is constructive, and shows moreover that the diameter of the Cayley graph using this generating set is at most $$n^2$$. In other words, every element of the group can be written as a product of elementary matrices of the first kind using a product length of at most $$n^2$$.

Getting a smaller generating set
Further, since $$E_{ij}(\lambda)E_{ij}(\mu) = E_{ij}(\lambda + \mu)$$, it suffices to restrict $$\lambda$$ to a subset of $$K$$ that generates $$K$$ additively. Note, however, that if we choose this smaller generating set, the diameter of the Cayley graph could be higher.

Finite fields
We consider the case of a field of size $$q = p^r$$ with characteristic $$p$$, and we are interested in studying $$SL(n,q)$$.

Related facts

 * Transpositions generate the finitary symmetric group
 * 3-cycles generate the finitary alternating group

Generalizations to beyond fields

 * Elementary matrices of the first kind generate the special linear group over a local ring
 * Elementary matrices of the first kind generate the special linear group over a Euclidean ring

Proof
The proof relies on a slight variation of the usual Gauss-Jordan elimination procedure. The variation is necessary because we do not have access to the other types of elementary matrices (permutations and scalar multiplications).