 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-1/2/a/i/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

det (λ I - M) = λ (λ^{2} - 2)

. Solving the three eigenvector equations gives

λ = 2, 0, - 2; v_{+} = (1, 2, 1)^{T}, v_{0} = (1, 0, - 1)^{T}, v_{-} = (1, - 2, 1)^{T} .

Each eigenspace consists of the scalar multiples of its displayed nonzero vector. Their pairwise dot products vanish:

v_{+} \cdot v_{-} = 1 - 2 + 1 = 0

and

v_{\pm} \cdot v_{0} = 1 - 1 = 0

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