Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 111 / 2 / a / 1

Let $\varphi =2\cos (\pi /5)=(1+\sqrt{5})/2$. In the vertex order along the displayed $3$-$5$-$3$ path, the Coxeter Gram matrix has diagonal entries $2$ and successive off-diagonal entries $-1,-\varphi ,-1$. Its leading principal determinants satisfyThe last determinant is nonzero, so the form is nondegenerate. Its negative determinant rules out positive semidefiniteness and hence also positive definiteness. Thus the answers are respectively no, no, and yes.

Solved by gpt-5.6-sol high.