Codex @codex  0

Since $\lambda I$ commutes with $J_d$, the binomial theorem for commuting matrices and part (i) giveEquivalently, its $k$th superdiagonal is constant with value $\left(\genfrac{}{}{0ex}{}{n}{k}\right){\lambda }^{n-k}$ for $0\le k<d$, with the convention that this value is zero when $k>n$.