Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 2 / 8 / b

The transpose reverses a matrix product. Since $S^T=S$, we have $(S^2{)}^T=S^T{S}^T=S^2$. Therefore $S^2$ is a symmetric matrix.

Apply trace orthogonality of symmetric and antisymmetric matrices. In detail, invariance of the trace under transpose and its cyclic property giveTherefore $\boxed{\mathrm{tr}(SA)=0}$. The negative sign comes from $A^T=-A$, while $S^T=S$.