Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 3 / Solution

The vectors $(1,1{)}^{\mathsf{T}}$ and $(1,-1{)}^{\mathsf{T}}$ are eigenvectors: multiplication adds or subtracts the two entries in a row. Their eigenvalues are thereforeThe second eigenvalue may be positive, zero or negative as the real parameter changes.