Jakob Nielsen (1924) showed that the automorphisms defined by the elementary Nielsen transformations generate the full automorphism group of a finitely generated free group.
The automorphism group of the free group with ordered basis [ x1, …, xn ] is generated by the following 4 elementary Nielsen transformations: These transformations are the analogues of the elementary row operations.
Transformations of the third kind correspond to scaling a row by an invertible scalar.
Transformations of the fourth kind correspond to row additions.
Nielsen gave a rather complicated finite presentation using these generators, described in (Magnus, Karrass & Solitar 2004, p. 165, Section 3.5).