Past exam of the mathematical methods I course of the University of Oxford / 2014 / 12 / iii / Solution

The cyclic property of the trace gives $\mathrm{tr}(AB)=\mathrm{tr}(BA)$, which is C1. The linearity of trace and matrix multiplication gives C2 and C3. Finally symmetry of $A$ giveswith equality exactly when every entry is zero. Thus C4 also holds. This is the Frobenius inner product restricted to the real vector space of symmetric matrices.