Inverse of Proper Orthogonal Matrix is Proper Orthogonal
Jump to navigation
Jump to search
Theorem
Let $\mathbf A$ be a proper orthogonal matrix.
Then its inverse $\mathbf A^{-1}$ is also proper orthogonal.
Proof
By definition of proper orthogonal matrix:
- $\mathbf A$ is orthogonal
- the determinant of $\mathbf A$ is equal to $1$.
By Inverse of Orthogonal Matrix is Orthogonal, $\mathbf A^{-1}$ is orthogonal.
By Determinant of Inverse Matrix:
- $\det \mathbf A^{-1} = \dfrac 1 {\det \mathbf A} = 1$
Hence, by definition, $\mathbf A^{-1}$ is proper orthogonal.
$\blacksquare$