1. The problem statement, all variables and given/known data I have to find an orthogonal matrix with an eigenvalue that does not equal 1 or -1. That's it. I'm completely stumped. 2. Relevant equations An orthogonal matrix is defined as a matrix whose columns are an orthonormal basis, that is they are all orthogonal to each other and each vector has length 1. These matrices have the property that their inverse is the same as their transpose. I don't think their are any other equations. 3. The attempt at a solution My professor claims that this is possible. So far I thought about a 1x1 matrix, as that is defined as each vector being orthogonal to each other, but the vector only has length 1 if the matrix is  or [-1]. And rectangular matrices don't have inverses. I'm stumped.