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Geometric action of an arbitrary orthogonal 3x3 matrix with determinant -1


Oct 7, 2013
I have a question about describing geometrically the action of an arbitrary orthogonal 3x3 matrix with determinant -1. I would like to know if my proposed solutions are satisfactory, or if they lack justification. I have two alternate solutions, but have little confidence in their validity. Any help would be greatly appreciated!!

Solution 1: The orthogonal 3 x 3 matrix with determinant −1 is an improper rotation, meaning it is a reflection combined with a proper rotation. In another sense, an improper rotation is an indirect isometry, which is an affine transformation with an orthogonal matrix with a determinant −1.

Solution 2: A rotation about the origin, followed by inversion through the origin (i.e. (x,y,z)-->(-x,-y,-z) ). Note that a "left-handed object" turns into a "right handed object", so "handedness is reversed" but otherwise it is just like a rotation.

Thanks in advance!


Well-known member
MHB Math Helper
Jan 29, 2012
Your "solution 1" does not say "rotation about the origin" so "solution 2" is better. Other than that, they both say the same thing.