Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 1E (Linear Algebra)

Let $E_{rs}$ be the matrix unit with its only nonzero entry in row $r$, column $s$. Forone hasThereforeThe $n^2$ matrix units form an eigenbasis, with $E_{rs}$ having eigenvalue $r-s$. The zero eigenspace consists exactly of the diagonal matrices and has dimension $n$. The rank-nullity theorem now gives

Fordirect multiplication givesWith respect to the ordered standard basis $(E_{11},E_{12},E_{21},E_{22})$, its matrix is therefore

This nilpotent operator satisfies ${\varphi }_A^3=0$, while ${\varphi }_A^2\ne 0$. Moreover,These ranks determine one nilpotent block of size three and one of size one. Hence its Jordan normal form is