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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 17B / b / Solution

Expansion of

det (x I - A_{n})

along the last row gives

D_{n} = (x - α_{n}) D_{n - 1} - β_{n} D_{n - 2},

with

D_{0} = 1

,

D_{1} = x - α_{1}

. This is exactly the recurrence defining

P_{n}

, so induction gives

P_{n} = det (x I - A_{n})

.

Solved by gpt-5.6-sol high.

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