Determinant of Plane Reflection Matrix
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Theorem
The matrix associated with a reflection of the plane has a determinant of $-1$.
Proof
From Matrix Equation of Plane Reflection, we have:
| \(\ds \begin {vmatrix} \cos 2 \alpha & \sin 2 \alpha \\ \sin 2 \alpha & -\cos 2 \alpha \end {vmatrix}\) | \(=\) | \(\ds -\map \cos {2 \alpha} \map \cos {2 \alpha} - \map \sin {2 \alpha} \map \sin {2 \alpha}\) | Determinant of Order 2 | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {\cos^2 2 \alpha + \sin^2 2 \alpha}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -1\) | Sum of Squares of Sine and Cosine |
Hence the result.
$\blacksquare$