Show that $O_2(\mathbb{R})$ contains only rotational and reflective symmetries. Show that $O_2(\mathbb{R})$ contains only rotational and reflective matrices.
I know that rotational and reflective symmetries are part of $O_2(\mathbb{R})$. I want to show that there is no other matrix that satisfies $O_2(\mathbb{R})$. This is what I have and I don't know what to do after this or if I'm going about this in the right way at all.
Suppose there exists a matrix $A \in O_2(\mathbb{R})$ such that is not a rotational or reflexive symmetry.
$A^T = A^{-1}$ by definition
Therefore we get the following simultaneous equations:
$$a = \frac{d}{ad - bc}; d = \frac{a}{ad - bc}; b = \frac{-c}{ad - bc}; c = \frac{-b}{ad - bc}$$
using the first two equations, you get $ad - bc =1$ or $a = -d$ and from the second two $ad - bc = -1$ or $b = -c$.
I don't know where to go from here to show that $a,b,c,d$ form either a rotational or reflective  matrix.
 A: By definition $A \in O_2(\mathbb{R})$ iff $AA^T = A^TA = I$.
Let
\begin{align}
A = 
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\end{align}
then we have that
\begin{align}
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\begin{pmatrix}
a & c\\
b & d
\end{pmatrix}
=
\begin{pmatrix}
a^2+b^2 & ac+bd\\
ac+bd & c^2+d^2
\end{pmatrix}
=
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}.
\end{align}
Hence, $(a, b)$ and $(c, d)$ are unit vectors. Moreover, $(a, b)$ is orthogonal $(c, d)$ since $ac+bd=0$. Then we can rewrite $A$ as follows
\begin{align}
A = 
\begin{pmatrix}
a & -b\\
b & a
\end{pmatrix} \ \ \text{ or } \ \ A = 
\begin{pmatrix}
a & b\\
b & -a
\end{pmatrix}
=
\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}
\begin{pmatrix}
b & -a\\
a & b
\end{pmatrix}
\end{align}
Since $a^2+b^2=1$, we can reparametrize the matrix by noting that $\cos^2\theta+\sin^2\theta = 1$. Hence, we have that either
\begin{align}
A = 
\begin{pmatrix}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{pmatrix}
\ \ \text{ or } \ \ 
A = 
\begin{pmatrix}
\cos\theta & \sin\theta\\
\sin\theta & -\cos\theta
\end{pmatrix}
=
\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}
\begin{pmatrix}
\sin\theta & -\cos\theta\\
\cos\theta & \sin\theta
\end{pmatrix}.
\end{align}
