Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 6A / c / i / Solution

The determinant $\det (M-tI)$ is a polynomial in $t$ with leading term $(-t{)}^n$, so it is not the zero polynomial and has only finitely many roots. Hence one may choose $\alpha >0$ smaller than every positive root, if any. Then

Solved by gpt-5.6-sol high.