Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 8 / c / ii

For the quadratic form of a squared cross product, express the two scalar terms as quadratic forms in $\mathbf{a}$:Then ${\mathbf{a}}^{T}M\mathbf{a}=|\mathbf{b}{|}^2|\mathbf{a}{|}^2-(\mathbf{a}\cdot \mathbf{b}{)}^2$. In particular, for $\mathbf{b}=(b_1,b_2,0)$,The cancellation is structural: $M\mathbf{b}=0$, while its action on every vector perpendicular to a nonzero $\mathbf{b}$ is multiplication by $|\mathbf{b}{|}^2$. Its eigenvalues are therefore $0,|\mathbf{b}{|}^2,|\mathbf{b}{|}^2$; for $\mathbf{b}=0$ the whole matrix is zero.