Can the product of some matrices equal the identity matrix?

Let $s=\begin{bmatrix}0 & -1 \\1&0\end{bmatrix}$ and $t=\begin{bmatrix}1 & -1 \\ 1&0\end{bmatrix}$. Then, in $PSL_2(\mathbb{Z})$, $s^2=t^3=1$, $s,t$ generate this group and have no other nontrivial relation (see https://chiasme.wordpress.com/2015/03/08/an-elementary-application-of-ping-pong-lemma/).

Now, note that in $PSL_2(\mathbb{Z})$, $(ts)^{2}=A$ while $B=(st)^2$.

So we want to see that no word in $stst$ and $tsts$ simplifies to the trivial word when $s^2=t^3=1$. That looks obvious (we can only reduce when $stst$ comes after $tsts$ and then the result is $tst^{-1}st$ which cannot be reduced further).