Generating set for $SL_2(\mathbb{C})$ Let
\begin{equation}
A = 
\begin{pmatrix}
\lambda & 0\\
0 & \frac{1}{\lambda}\\
\end{pmatrix}
\end{equation}
be a matrix in $SL_2(\mathbb{C}$). I am trying to prove that $SL_2(\mathbb{C})$ is generated by the matrices
\begin{equation}
\begin{pmatrix}
1 & z\\
0 & 1\\
\end{pmatrix},
\begin{pmatrix}
1 & 0\\
z & 1\\
\end{pmatrix}
\end{equation}
for $z\in\mathbb{C}$. I have already handled all the cases expect for matrices like $A$. How can I write it as a product of matrices of this type? Any hints are appreciated.

How about this:
$A = \begin{pmatrix}
1 & -\lambda \\
0 & 1
\end{pmatrix} \cdot \begin{pmatrix}
1 & 0\\
\frac{1}{\lambda} & 1\\
\end{pmatrix} \cdot \begin{pmatrix}
1 & 0\\
-1 & 1\\
\end{pmatrix} \cdot \begin{pmatrix}
1 & 1\\
0 & 1\\
\end{pmatrix} \cdot \begin{pmatrix}
1 & 0\\
\lambda - 1 & 1\\
\end{pmatrix}$
