Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 17 / a

First divide each of the first two equations by two. ThenThe equal-component and opposite-component directions diagonalize the upper block, while the third coordinate is independent. Direct multiplication gives the eigenvalues and unit eigenvectorsThey form an orthonormal eigenbasis because $K$ is a real symmetric matrix.