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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/2017/paper-2/17/a/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 / 2017 / Paper 2 / 17 / a / Solution by

Codex  0 2026-10-18

First divide each of the first two equations by two. Then

K = 3/2 - 1/2 0 - 1/2 3/2 0 004 .

(1)

The equal-component and opposite-component directions diagonalize the upper block, while the third coordinate is independent. Direct multiplication gives the eigenvalues and unit eigenvectors

γ 124 normalized eigenvector \frac{1}{2} (1, 1, 0)^{T} \frac{1}{2} (1, - 1, 0)^{T} (0, 0, 1)^{T} .

(2)

They form an orthonormal eigenbasis because

K

is a real symmetric matrix.

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