 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/13/a/ii/solution

For either sign, the dot product with

v_{1}

v_{1} \cdot v_{\pm} = (- 5 \pm 15) + (2 \mp 15) + 3 = 0.

The remaining pair also satisfies

v_{+} \cdot v_{-} = (25 - 15) + (4 - 15) + 1 = 0 .

Hence the three eigenvectors are mutually orthogonal. This also follows from the spectral theorem for real symmetric matrices, since

M = M^{T}

and its eigenvalues are distinct, but the displayed products verify the actual chosen vectors directly.

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