Elementary Row Matrices Let $A$ = 
$$
\begin{align}
\begin{bmatrix}
-4 & 3\\
1 & 0
\end{bmatrix}
\end{align}
$$
Find $2 \times 2$ elementary matrices $E_1$,$E_2$,$E_3$ such that $A$ = $E_1 E_2 E_3$
I figured out the operations which need to be performed which are;
$E_1$ = $R_2 \leftrightarrow R_1$
$E_2$ = $R_2$ = $R_2$ + $4R_1$
$E_3$ = $R_2$ * $\frac{1}{3}$
My question is how would I go about writing the elementary matrices? The solution says that they are;
$E_1$ =
$
\begin{align}
\begin{bmatrix}
1 & -4\\
0 & 1
\end{bmatrix}
\end{align}
$
$E_2$ =
$
\begin{align}
\begin{bmatrix}
3 & 0\\
0 & 1
\end{bmatrix}
\end{align}
$
$E_3$ =
$
\begin{align}
\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}
\end{align}
$

Note that the solutions are not unique. With your elementary row operations, we have
$$
\pmatrix{1&0\\ 0&\tfrac13}\pmatrix{1&0\\ 4&1}\pmatrix{0&1\\ 1&0}A = I_2.
$$
Therefore, by performing the reverse row operations (and also in reverse order) on $I_2$, we get
$$
A = \pmatrix{0&1\\ 1&0}\pmatrix{1&0\\ -4&1}\pmatrix{1&0\\ 0&3}I_2.
$$
