Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 19I / ii

The pairing is real and symmetric: $\overline{\mathrm{tr}(AB)}=\mathrm{tr}(B^{\ast }{A}^{\ast })=\mathrm{tr}(BA)=\mathrm{tr}(AB)$, and cyclicity of the trace exchanges $A,B$. For the coordinates in part (i), $A^2=-(x^2+y^2+z^2)I$, soThus it is a positive-definite real bilinear form, hence nondegenerate. Finallyproving invariance under conjugation.