Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 2 / 14 / b / i

The transpose reverses product order, so $(CB{)}^{\mathsf{T}}=B^{\mathsf{T}}{C}^{\mathsf{T}}$. If $B$ is an orthogonal matrix, then $B^{-1}=B^{\mathsf{T}}$ and the given identity becomesMultiply on the left by $B$ to obtain $C^{\mathsf{T}}+C=0$. Therefore $C$ is antisymmetric, so it is an antisymmetric matrix, namely $\boxed{C^{\mathsf{T}}=-C}$.