We know that every matrix in $\mathrm{GL}(2,\mathbb C)$ is a scalar multiple because of the isomorphism between $\mathrm{GL}(2,\mathbb C)/(\mathbb{C}^*)I_2$.
There are four kinds of products of matrices:
$$\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix},$$ where $k>0$;
$$\begin{pmatrix} \lambda & 0 \\ 0 & 1 \end{pmatrix},$$ where $\lambda = 1$;
$$\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix},$$ where $b$ is inclusive of all complex numbers;
$$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$$
I am a little confused how to utilize these products to prove that every matrix in $\mathrm{GL}(2,\mathbb C)$ is a product of matrices of these four kinds