Prove that all 2x2 orthogonal matrices can be expressed as Rotation or Reflection

Solution 1:

With not so many variables running around, we may verify this claim algebraically. Write $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Then $A^TA = \begin{pmatrix} a^2+c^2 & ab+cd \\ ab+cd & b^2+d^2 \end{pmatrix}$. So $\begin{pmatrix} a \\ c\end{pmatrix}$ and $\begin{pmatrix} b \\ d\end{pmatrix}$ are unit vectors so that $\begin{pmatrix} a \\ c\end{pmatrix}\cdot \begin{pmatrix} b \\ d\end{pmatrix} = 0$. We can check that this implies $b = \pm c$ while $d = \mp a$. So $A = \begin{pmatrix} a & c \\ c & -a \end{pmatrix}$ or $A = \begin{pmatrix} a & -c \\ c & a \end{pmatrix}$, which are the forms of a reflection composed with a rotation and a rotation, respectively.