Answer
$A=\begin{bmatrix}
1&0\\
1& 1
\end{bmatrix}.\begin{bmatrix}
1&2\\
0& 1
\end{bmatrix}$
Work Step by Step
We can reduce A to row-echelon form using the following sequence of elementary row operations:
$\begin{bmatrix}
1&2\\
1& 3
\end{bmatrix} \approx^1\begin{bmatrix}
1&2\\
0& 1
\end{bmatrix} \approx^2\begin{bmatrix}
1&0\\
0& 1
\end{bmatrix}$
$1.A_{12}(-1)$
$2.A_{21}(-2)$
So, elementary matrices are $A_{21}(-2),A_{12}(-1)$
or $E_1=\begin{bmatrix}
1&-2\\
0& 1
\end{bmatrix},E_2=\begin{bmatrix}
1&0\\
-1& 1
\end{bmatrix}$
$A=E_1E_2A=I_2$
$\rightarrow A=E_2^{-1}E_1^{-1}=\begin{bmatrix}
1&0\\
1& 1
\end{bmatrix}.\begin{bmatrix}
1&2\\
0& 1
\end{bmatrix}$
