I'm given two matrices in $SL_2(\mathbb{Z})$

$
A = \left(\begin{array}{cc}
       2 & 3\\
       3 & 5
     \end{array}\right), B = \left(\begin{array}{cc}
       5 & 3\\
       3 & 2
     \end{array}\right) $

Then the group $<A,B>$ is free of rank 2. Now my problem is to prove that the following matrix:

$\left(\begin{array}{cc}
       1 & n\\
       0 & 1
     \end{array}\right),n\in \mathbb{Z^*}$

does not belong to $<A,B>$. That is cannot be obtained from products of $A$ and $B$ and their inverses.

I started by writing
$ \left(\begin{array}{cc}
       0 & 1\\
       - 1 & 0
     \end{array}\right) \longrightarrow x, \left(\begin{array}{cc}
       0 & 1\\
       - 1 & 1
     \end{array}\right) \longrightarrow y,
A \longrightarrow (y x y^{- 1} x^{-1})^2,
B \longrightarrow (y^2 x y^2 x^{- 1})^2
$
But then did not get any further. 

Any help is highly appreciated.