Linear Algebra 21e: What All 2x2 Orthogonal Matrices Look Like

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[@MathTheBeautiful]

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[@menturinai1387]

You left as an exercise at the end of the video to left-multiply the standard rotation matrix by a to-be-determined elementary matrix. I am unable to find the matrix that would go there that is not terms of cosines and sines. We need to get rid of the negative sign in the 12 element, so we make 11 element equal to -1, but this makes the 11 element negative as well, which cannot be undone. Swapping rows won't help either since we need to keep the cosine in the 11 spot. Is there something I'm missing?

[@TheViscweaver]

Scratched my head about this very issue. The only resolution I could find is: After every flip the protocol that counter-clockwise rotation is considered a positive angle gets reversed. That is to say a flip which transforms the space into a mirror image also changes what looks like counter-clockwise into clockwise rotation. So after a flip a counter-clockwise rotation is a rotation with a -ve sign.

[@cjpwdesign6108]

the solution would be to premultiply the transposed rotation matrix by the elementary reflection matrix about the x-axis

[@AshiRajput-qf8gp]

We can take 4 by 4 order in orthogonal matrix??? If yes then prove