Expressing elementary matrices in terms of each another How can I express an elementary matrix of type 2 in terms of the product of elementary matrices of types 1 and 3? Just for clarity, here are the types:
Type 1: \begin{bmatrix}1&a\\0&1\\ \end{bmatrix} or 
\begin{bmatrix}1&0 \\a&1\end{bmatrix}
Type 2: \begin{bmatrix}0&1\\1&0\\ \end{bmatrix}
Type 3: \begin{bmatrix}c&0\\0&1\\ \end{bmatrix} or 
        \begin{bmatrix}1&0\\0&c\\ \end{bmatrix}
I have been trying to do this, but I can only think of trial and error or brute-forcing. Is there a proof-based way to do this? Thanks in advance.

It seems that you want to be able to swap two rows using only row replacement and scaling operations. By row reducing your Type 2 matrix to the identity (without swapping!), we obtain:
$$
\begin{bmatrix}0&1\\1&0\\ \end{bmatrix}
\sim
\begin{bmatrix}1&1\\1&0\\ \end{bmatrix}
\sim
\begin{bmatrix}1&1\\0&-1\\ \end{bmatrix}
\sim
\begin{bmatrix}1&0\\0&-1\\ \end{bmatrix}
\sim
\begin{bmatrix}1&0\\0&1\\ \end{bmatrix}
$$
where we did:


*

*Add $1 \cdot (\text{Row }2)$ to $\text{Row 1}$.

*Add $-1 \cdot (\text{Row }1)$ to $\text{Row 2}$.

*Add $1 \cdot (\text{Row }2)$ to $\text{Row 1}$.

*Scale $(\text{Row }2)$ by $-1$.


Translating these row operations to elementary matrices and working backwards, we obtain:
$$
\begin{bmatrix}0&1\\1&0\\ \end{bmatrix}
=
\begin{bmatrix}1&0\\0&-1\\ \end{bmatrix}
\begin{bmatrix}1&1\\0&1\\ \end{bmatrix}
\begin{bmatrix}1&0\\-1&1\\ \end{bmatrix}
\begin{bmatrix}1&1\\0&1\\ \end{bmatrix}
$$
