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 / i / Solution

With $\nu =0$, the characteristic polynomial factors into $(1-\lambda )[{\lambda }^2-\lambda -({\mu }^2+4)]$, up to the overall sign convention. The eigenvalues are $1$ and $(1\pm \sqrt{17+4{\mu }^2})/2$. Integer values require an odd integer $m$ satisfying $(m-2\mu )(m+2\mu )=17$. The positive integer factorization $17=1\cdot 17$ forces $m=9$ andThe resulting eigenvalues and corresponding nonzero eigenvectors areAny nonzero scalar multiple is equally valid. The factorization also shows that no other positive integer $\mu$ works.