Expressing elementary matrices in terms of each another

Question

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.

Answer 1

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:

1. Add $1 \cdot (\text{Row }2)$ to $\text{Row 1}$.
2. Add $-1 \cdot (\text{Row }1)$ to $\text{Row 2}$.
3. Add $1 \cdot (\text{Row }2)$ to $\text{Row 1}$.
4. Scale $(\text{Row }2)$ by $-1$.

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