Past exam of the linear algebra course of the University of Oxford / 2017 / 1 / b / ii / Solution

Write $B=D+K$, where $D$ has $A$ in each diagonal block and zero elsewhere, while $K$ has $I$ immediately above the diagonal and zero elsewhere. These block matrices satisfy $DK=KD$: either product has $A$ immediately above the diagonal. Also, $K^i$ has $I$ on its $i$th superdiagonal and is zero elsewhere.

The binomial theorem for commuting matrices therefore yieldsOnly $r=i$ can contribute to block $(k,k+i)$, provided that block lies in the matrix. HenceThus ${\beta }_{i,j}=\left(\genfrac{}{}{0ex}{}{j}{i}\right)$, interpreted as an element of the field. The formula remains valid in positive field characteristic: some of these coefficients can then vanish. This possibility matters in the next part.