Suppose we have integer matrices $A\_1,\ldots,A\_n\in\operatorname{GL}(n,\mathbb Z)$. Define $\varphi:F\_n\to\operatorname{GL}(n,\mathbb Z)$ by $x\_i\mapsto A\_i$. Is there an algorithm to decide whether or not $\varphi$ is injective?



Is it decidable whether or not a collection of integer matrices generates a free group?
For $n=1, 2$ the answer is "yes" since the group is virtually free, for $n\ge 3$ the answer is not known (an open problem). 


 **Edit.**  In fact even for two $n\times n$-matrices the problem is open. Moreover the solution of the following ``easier" problem is not known: for which algebraic integers $\lambda$ the matrices $\left(\begin{array}{ll} 1 & 2\\\ 0 & 1 \end{array}\right)$ and $\left(\begin{array}{ll} 1 & 0\\\ \lambda & 1 \end{array}\right)$ generate a free group (see  [this](http://cage.ugent.be/~bamberg/Research_files/Non-free%20points%20for%20groups%20generated%20by%20a%20pair%20of%202x2%20matrices.pdf)  paper, for example). The fact that this problem is easier follows from the trivial observation that the group generated by these two matrices is isomorphic to some effectively computable group of $n\times n$-integer matrices for some $n\ge 2$ (depending on the degree of the algebraic number $\lambda$). 


