It is a classical fact that free (discrete) groups can be embedded in $GL_{2}(\mathbb{Z})$. In 1987, Zubkov showed that for a free pro-$p$ group $F_{\hat{p}}$, the situation changes, and when $p>2$, $F_{\hat{p}}$ cannot be embedded in $GL_{2}(\Delta)$ when $\Delta$ is a profinite ring. In 2005, inspired by Kemer's solution to the Specht problem, Zelmanov sketched a proof for the following generalization: For every $d\in\mathbb{N}$ and large enough prime $p\gg d$, $F_{\hat{p}}$ cannot be embedded in $GL_{d}(\Delta)$. The natural question then is: What can be said when $p$ is not large enough? What can be said in the case $d=p=2$ ? In the talk I am going to describe the proof of the following theorem: $F_{\hat{2}}$ cannot be embedded in $GL_{2}(\Delta)$ when $char(\Delta)=2$. The main idea of the proof is the use of trace identities in order to apply finiteness properties of a Noetherian trace ring through the Artin-Rees Lemma (Joint with E. Zelmanov).