Let $m$ be a positive integer, and let $A_1,B_1 \in \operatorname{SL}(2,\mathbb{Z})$. Can one always find matrices $A_2,B_2 \in \operatorname{SL}(2,\mathbb{Z})$ such that $$ A_1 \left( \begin{array}{cc} m^{-1} & 0 \\ 0& m\end{array} \right)B_1 =A_2 \left( \begin{array}{cc} m & 0 \\ 0& m^{-1}\end{array} \right)B_2 $$ PS: $\operatorname{SL}(2,\mathbb{Z})$ are the set of all $2\times 2$ matrices $A$ with entries in $\mathbb{Z}$ and $\det(A)=1$.
Many thanks.
Hint: let $J= \left( \begin{array}{cc} 0 & 1 \\ -1& 0\end{array} \right)$. Note that $J^2=-I$ and $$J \left( \begin{array}{cc} m^{-1} & 0 \\ 0& m\end{array} \right)(-J)= \left( \begin{array}{cc} m & 0 \\ 0& m^{-1}\end{array} \right)$$