Is it possible to find two matrices $A$ and $B$, so that there does not exists a product of matrices $A$,$A^{-1}$,$B$,$B^{-1}$ that is equal to $Id$, under the condition that the product is irreducible, that is, it is is not trivial as a word (e.g. $AA^ {-1}B^{-1}B$).
If there is no simple answer I would be happy with some references to where I should start.



Trivial product of two matrices?
If I understand correctly, you will find the answer in a good exposition of the [Banach-Tarski paradox](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#A_sketch_of_the_proof). Finding two rotation matrices in $\mathbb{R}^3$ that generate the free group in two generators is a usually the biggest part of the construction.


