For a $3 \times 3$ matrix $A$, say that a point $p$ on the unit sphere centred at the origin in $\mathbb{R}^3$ is a pole of $A$ if $Ap = p$. Denote by $\mathrm{SO}_3$ the subgroup of $\mathrm{GL}_3(\mathbb{R})$ consisting of all the orthogonal matrices with determinant 1. (A) Show that if $A \in \mathrm{SO}_3$, then $A$ has a pole. (B) Let $G$ be a subgroup of $\mathrm{SO}_3$. Show that $G$ acts on the set $$\left\{p \in \mathbb{S}^2 \mid p \text{ is a pole for some matrix } A \in G\right\}.$$
For a $3 \times 3$ matrix $A$, say that a point $p$ on the unit sphere centred at the origin in $\mathbb{R}^3$ is a pole of $A$ if $Ap = p$. Denote by $\mathrm{SO}_3$ the subgroup of $\mathrm{GL}_3(\mathbb{R})$ consisting of all the orthogonal matrices with determinant 1.\\
(A) Show that if $A \in \mathrm{SO}_3$, then $A$ has a pole.\\
(B) Let $G$ be a subgroup of $\mathrm{SO}_3$. Show that $G$ acts on the set
$$\left\{p \in \mathbb{S}^2 \mid p \text{ is a pole for some matrix } A \in G\right\}.$$