How many matrices does SO(2) contain?

Question

Would I be correct in saying that the special orthogonal group SO(2) contains one matrix , namely;

$A=\begin{pmatrix}{} \cos\theta& -\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}$

or would it have infinitely many matrices as $\theta $ can be any angle between 0 and 360, and different angles would produce different entries ?

Answer 1

Infinitely many matrices: $$ {\rm SO}(2,\Bbb R) = \left\{A=\begin{pmatrix}{} \cos\theta& -\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}\mid \theta \in \Bbb R \right\}. $$For example, for $\theta = 0$ and $\theta = \pi/2$ we get $$\begin{pmatrix}{} 1 & 0\\ 0 & 1 \end{pmatrix}, \begin{pmatrix}{} 0 & -1\\ 1 & 0 \end{pmatrix} \in {\rm SO}(2,\Bbb R).$$