Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 7 / Solution

This is a generalized eigenvalue problem. A nonzero vector lies in the kernel of $A-\lambda B$ precisely when its determinant vanishes:HenceThe corresponding eigenvectors can be taken as $(1,-1{)}^T$ and $(1,1{)}^T$, respectively, which also checks that both values give nontrivial solutions.