Matrices as generators of free group.

Question

In the introduction section of the paper Triples of $2\times 2$ matrices which generate free groups the authors mentioning some thing...

In my words:

The matrices $\begin{pmatrix}1 & 0 \\ 2 & 1\end{pmatrix}$ and $\begin{pmatrix}1 & 2 \\ 0 & 1\end{pmatrix}$ are generating the free group of two generators.

How to prove the above statement?

Answer 1

Your question is a particular case of Theorem 14.2.1 in Kargapolov and Merzljakov, "Fundamentals of the Theory of Groups", but you can easily adapt their proof to your case.

Answer 2

Answer is in the this paper of Lyndon and Ullman.

Answer 3

This is nothing more than an application of the Ping-Pong Lemma.