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determinant of a rotation matrix

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chaospixel

determinant of a rotation matrix
September 11, 2006 10:04PM

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Why must the determinant of a rotation matrix be 1? I get that its inverse is its transpose and therefore it is orthonormal. but I don't get why that means its determinant is 1.

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chaospixel

Re: determinant of a rotation matrix
September 11, 2006 11:15PM

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The best I seem to be able to find online is that a rotation matrix is an orthogonal matrix with a determinant of 1. I also found out that any orthogonal matrix must have a determinant of either 1 or -1 because:

1 = det(I) = det(Q^T Q) = det(Q^T)det(Q) = (det(Q))^2 so det(Q) = +/-sqrt(1)

does anyone see _why_ an orthogonal matrix with a determinant of 1 _must_ rotate some things around some point by some angle?