Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 2 / 9 / a

A real normal matrix satisfies $M{M}^T=M^{T}M$. For a symmetric matrix, $M^T=M$ and both products equal $M^2$. For an antisymmetric matrix, $M^T=-M$ and both equal $-M^2$. For an orthogonal matrix, both products are the identity. Thus each of the three classes consists of normal matrices.