On a finite state representation of $GL(n,\mathbb{Z})$

It is examined finite state automorphisms of regular rooted trees constructed to represent groups $GL(n,\mathbb{Z})$. The number of states of automorphisms that correspond to elementary matrices is computed. Using the representation of $GL(2,\mathbb{Z})$ over an alphabet of size $4$ a finite state representation of the free group of rank $2$ over binary alphabet is constructed.