Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 13E / iii / Solution

An example in dimension four isBoth are invertible, so each has rank four. Both have characteristic polynomial $(x-1{)}^4$, and each has minimal polynomial $(x-1{)}^2$, since its nonzero nilpotent part squares to zero. However $\mathrm{rank}(A-I)=2$ and $\mathrm{rank}(B-I)=1$. If $B=PA{P}^{-1}$, then $B-I=P(A-I)P^{-1}$, which would preserve these ranks. The contradiction proves that the matrices are not similar, despite all three specified invariants agreeing.