Let $$GL_2 = \left\{ A\in M_2(\mathbb{Z}\mid \det A \in \{\pm 1\} \right\}$$
$$a = \left(\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array}\right), b = \left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right)$$ Show using the Euclid's algorithm for GCD that for every $g\in GL_2$ there are $s_1,\ldots,s_k, r_1,\ldots,r_k$ such that $$g = a^{r_1}\cdots a^{r_k}b^{s_1}\cdots b^{s_k}$$
So we learned in class about the matrix method for Euclid's algorithm. Also, I know that $\det A = ad-bc$ so in our case that means that $ad-bc = \pm 1$ which means that $\gcd (ad,bc) = 1$. In addition, I know that since $A$ is invertible then there are $E_1,\ldots,E_m$ such that $A = E_1\cdots E_m$.
I tried to connect the dots for those facts above, but couldn't put my finger on the solution.
I'd be glad for help!