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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ia/paper-1/6a/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 6A / d / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

P (A) = (- 1)^{n - 1} A^{n - 1} - c_{n - 1} A^{n - 2} - \dots - c_{2} A - c_{1} I .

The Cayley-Hamilton theorem for

χ_{A} (z) = (- 1)^{n} z^{n} + c_{n - 1} z^{n - 1} + \dots + c_{1} z + c_{0}

gives

A P (A) = P (A) A = c_{0} I = det (A) I .

If

A

is nonsingular, multiplication by

A^{- 1}

shows

P (A) = det (A) A^{- 1} = adj (A)

.

For arbitrary

A

, apply the nonsingular result to

A - t I

for a sequence of nonzero

t \to 0

avoiding the finitely many singular values. Both the characteristic coefficients and the adjugate entries depend polynomially on the matrix entries, so the limit gives

adj (A) = (- 1)^{n - 1} A^{n - 1} - c_{n - 1} A^{n - 2} - \dots - c_{2} A - c_{1} I .

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