Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 6 / c / Solution

The second coordinate direction is already an eigenvector with eigenvalue three. The first and third coordinates form a symmetric block, whose sum and difference directions giveFor example, the sum of the block's diagonal and off-diagonal entries is $(1+c)+(5-c)=6$, while their difference is $(1+c)-(5-c)=2c-4$. Equivalently the characteristic polynomial is $(\lambda -6)(\lambda -3)(\lambda -2c+4)$.

The displayed basis works for every real $c$. At $c=7/2$, the eigenvalue three has eigenspace spanned by the second and third listed vectors; at $c=5$, the eigenvalue six has eigenspace spanned by the first and third. Every nonzero vector in a repeated eigenspace is an eigenvector, and any orthonormal basis of it may replace the displayed choices.