TITLE: Show $SL(2,\mathbb{Z})$ written as finite product of elements of a particular form
QUESTION [7 upvotes]: Prove that any element of $SL(2,\mathbb{Z})$ can be represented by a finite product of matrices of the following form. $$\begin{pmatrix}1-ab & a^2\\ -b^2 & 1+ab\end{pmatrix}.$$ 
We are given that $SL(2,\mathbb{Z})$ is generated by $\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}$ and $ \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$. When $a=1$, $b=0$ we get the first generator, not sure how to find the second.
Its probably very easy but I am having trouble.

REPLY [3 votes]: So $a=1$, $b=0$ gives $X := \left(\begin{array}{cc}1&1\\0&1\end{array}\right)$, and $a=b=1$ gives $Y := \left(\begin{array}{cc}0&1\\-1&2\end{array}\right)$. 
Now $YX^2 = \left(\begin{array}{cc}0&1\\-1&0\end{array}\right)$, which is the inverse  of your second generator of ${\rm SL}(2,{\mathbb Z})$, but that has finite order, so that's OK.
Now, to solve the problem, we still need to get $X^{-1}$ as a product of matrices of the required form. But $X^{-1} = (YX^2)^{-1}YX = (YX^2)^3YX$.