Past exam of the mathematical methods I course of the University of Oxford / 2016 / 5 / Solution

The equations are an eigenvalue problem for the matrixA nonzero solution requires $\det (\lambda I-A)=0$. Expanding givesRow reduction gives the complete eigenspaces:For $\lambda =1$ the solutions form the plane $x-2y+z=0$ through the origin; for $\lambda =-3$ they form a line through the origin. Exclude the zero vector when requiring a nontrivial solution. The repeated eigenvalue $1$ has a two-dimensional eigenspace; there are three independent eigenvectors in total. This matrix is not symmetric, so its different eigenspaces need not be mutually orthogonal.