# Show $SL(2,\mathbb{Z})$ written as finite product of elements of a particular form

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.

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@scott: Are you allowed inverses of those matrices as well? – Jim Apr 24 '13 at 23:50
I don't believe that I am. – scott Apr 24 '13 at 23:56

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