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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 8E / a / ii / Solution

Since

λ I

commutes with

J_{d}

, the binomial theorem for commuting matrices and part (i) give

(λ I + J_{d})^{n} = k = 0 \sum m i n (n, d - 1) (k n) λ^{n - k} J_{d}^{k} .

Equivalently, its

k

th superdiagonal is constant with value

(k n) λ^{n - k}

for

0 \leq k < d

, with the convention that this value is zero when

k > n

.

Solved by gpt-5.6-sol high.

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