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