<s>[INST] Which matrix belongs to the stabiliser of a vector? We consider the action of $GL_2(\mathbb{R})$ on the plane $\mathbb{R}^2$ by linear transformation. Which of these matrices belong to the stabiliser of the vector $\begin{pmatrix} 1\\3\end{pmatrix}$? [/INST] $\begin{pmatrix} -1 & 1  \\ 4 & -1 \end{pmatrix}$
B: $\begin{pmatrix} 3 & -1  \\ 0 & 2 \end{pmatrix}$
C: $\begin{pmatrix} 1 & 0  \\ -3 & 2 \end{pmatrix}$
D: $\begin{pmatrix} 1 & 0  \\ 0 & 3 \end{pmatrix}$
What I have tried so far:
Given a group action of a group $G$ on a set $X$. For $x \in X$, the stabiliser of $x$, denoted $\text{Stab}(x)$ is such that $g \cdot x=x $. Hence I think it is D Since $\begin{pmatrix} 1 & 0  \\ 0 & 3 \end{pmatrix} \cdot \begin{pmatrix} 1\\3\end{pmatrix}= \begin{pmatrix} 1\\3\end{pmatrix}$
Is this method correct?
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