A reflection matrix is always of the form \[\begin{pmatrix} \cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & -\cos 2 \theta \end{pmatrix},\]where the vector being reflected over has direction vector $ \begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix} $. The determinant of this matrix is then \[(\cos 2 \theta)(-\cos 2 \theta) - \sin^2 2 \theta = -\cos^2 2 \theta - \sin^2 2 \theta = \boxed{-1}.\](Why does this make sense geometrically?)