I am a newcomer in Coding Theory and I am trying to solve the following question: \begin{equation*} \text{Let}\ \mathscr{C} \subset \mathbb{F}_3^4 \text{ be a code}, \text{with racio } 2/4, \text{and with the following encoder}, G_1=\begin{bmatrix} 1 & 1 & 1 & 0 \\ 2 & 0 & 1 & 1 \end{bmatrix}. \text{Find another encoder} \ G_2 \text{ of } \mathscr{C} \text{ such that } G_2=[I_2 \hspace{2mm} A] \text{ for some matrix }A∈ \mathbb{F^{2\times 2}}. \end{equation*}
$\textbf{This is my attempt to solve the question:}$
I know that that if $G_1 \in \mathbb{F}^{k \times n}$ is a encoder of $\mathscr{C}$, and $U \in \mathbb{F}^{k \times k}$ is an invertible matrix, and $G_2=U G_1$ then $G_2$ is also an encoder of $\mathscr{C}$. So I though I could just define a random matrix $U$, like, for example,
\begin{equation*}
U=\begin{bmatrix}
0 & 2 \\
1 & 1
\end{bmatrix},
\end{equation*}
such that $U$ is invertible (which is true because $\det(U)= -2 \neq 0$). And then do the multiplication $UG_1$ to find $A$:
\begin{equation*} G_2=UG_1 \Leftrightarrow \begin{bmatrix} 1 & 0 & a_{11} & a_{12} \\ 0 & 1 & a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 & 1 & 0 \\ 2 & 0 & 1 & 1 \end{bmatrix} \Leftrightarrow \ \end{equation*} \begin{equation*} \begin{bmatrix} 1 & 0 & a_{11} & a_{12} \\ 0 & 1 & a_{21} & a_{22} \end{bmatrix}= \begin{bmatrix} 1 & 0 & 2 & 2 \\ 0 & 1 & 2 & 1 \end{bmatrix} \end{equation*}
Therefore, $A=\begin{bmatrix} 2 & 2 \\ 2 & 1 \end{bmatrix}$, and we have another encoder for our code $\mathscr{C}$, $G_2=\begin{bmatrix} 1 & 0 & 2 & 2 \\ 0 & 1 & 2 & 1 \end{bmatrix}$.
$\textbf{Is my resolution correct?}$
Thank you so much in advance!