Use geometry
Fragment of a discussion from Science talk:Math Exam Resources/Courses/MATH152/April 2015/Question B 6 (a)
But an even more elegant solution for this particular problem is as follows (sketched below; please fill in the details):
A rotation matrix is orthogonal, and therefore has determinant ; it is easy to see in this case that the determinant is 1. One of the eigenvalues must be 1 (it's a rotation in an odd-dimensional space); let the two others be . Since the sum of the eigenvalues is the trace of the matrix, and since their product is the determinant, we have and , whence
Nicholas Hu (talk)