Groups and the Group Axioms · Hard problem

Let $A=\begin{pmatrix}2&1\\1&1\end{pmatrix}$ and $B=\begin{pmatrix}1&1\\0&1\end{pmatrix}$, invertible integer matrices with $A^{-1}=\begin{pmatrix}1&-1\\-1&2\end{pmatrix}$ and $B^{-1}=\begin{pmatrix}1&-1\\0&1\end{pmatrix}$. If $X$ satisfies $AXB=I$, find the $(1,1)$ entry of $X$.