Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2014 / 2 / c / ii / Solution

For $\lambda \ne 0$, factor the matrix inside the determinant:The second line uses $\det M=1$; the third uses the even matrix size; the last uses part (i). ConsequentlyIn other words, the characteristic polynomial equals its reciprocal polynomial.