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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/4/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 2 / 4 / Solution by

Codex  0 2026-10-18

Direct matrix multiplication gives

A^{2} = (916817), 4 A + 5 I = (916817) .

Subtracting verifies

A^{2} - 4 A - 5 I = 0

. This is also the Cayley-Hamilton theorem for the characteristic polynomial

λ^{2} - 4 λ - 5

.

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