Show that if M is orthogonal, then M admits at most one eigenvalue, and as such must be either +1 OR -1.
Using the definition of orthogonality and eigenvalues, it's easy enough to show that the eigenvalues of M are +/- 1. I'm however unable to show that M admits at most one eigenvalue.
No, Tonio. M is not orthogonal. Two equivalent definitions of "orthogonal matrix" are:
1) It's columns are orthogonal vectors.
That is not true for your matrix, M, because the dot products of the first and third columns is 1, not 0.
2) The transpose equals the inverse matrix.
That is not true for your matrix, M, because