Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 16 / b / ii / Solution

For $\nu =5$, $B$ is real symmetric. Sylvester's criterion says that all eigenvalues are positive exactly when its leading principal matrix minors are positive. These are $1$, $5-4=1$, and $\det B=1-{\mu }^2$. ThereforeAt either endpoint one eigenvalue is zero, so the strict positivity requirement excludes it.