1. The problem statement, all variables and given/known data Find all 2 x 2 and 3 x 3 orthogonal matrices which are diagonal. Construct an example of a 3 x 3 orthogonal matrix which is not diagonal. 2. Relevant equations Diagonal Matrix = All components are 0 except for the diagonal, for a 2x2 matrix, this would mean components a and d may or may not be 0, while b and c must be 0. Orthogonal IF: A^-1=A^T 3. The attempt at a solution So, if I have a diagonal 2x2 matrix, then it is automatically considered orthogonal for all values of a and d. I cannot think of a possibility where this doesn't hold. for a 3x3 matrix, this problem seems substantially more difficult as I have to calculate the inverse of the 3x3 diagonal matrix. The transpose should be easy to calculate. It is simply the same matrix. So, if the transpose of this 3x3 matrix has to equal the inverse, we don't need to do the inverse calculation. We can say all 3x3 matrices that are diagonal are also orthogonal. Now to construct an example of a 3x3 orthogonal matrix which is not diagonal, is a bit more difficult. I see that it has to be over the field of real numbers, but not getting further on this yet.