Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 8E / a / Solution

A direct determinant calculation givesMoreover,has rank one, so its kernel has dimension two, while $(A-I{)}^2=0$ and $A\ne I$. Hence the minimal polynomial isThere are two Jordan blocks, because the eigenspace has dimension two, and the largest has size two, because the minimal polynomial has exponent two. Thus

Solved by gpt-5.6-sol high.