Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 17 / a / Solution

A left eigenvector satisfies $(M-\lambda I{)}^T\mathbf{x}=0$. Here $M$ is a symmetric matrix, so its left and right eigenspaces have the same components. The characteristic polynomial factors asThe eigenvalues and all corresponding nonzero row eigenvectors areFor example, in the $xy$ block, equal components give the eigenvalue $3$ and opposite components give $-1$, while the $z$ coordinate is unaffected.