 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/18/b/iv/solution

Choose the unnormalized eigenvectors

u_{+} = (1, 1)^{T}, S u_{+} = 3 u_{+}, u_{-} = (1, - 1)^{T}, S u_{-} = - u_{-} .

x = \frac{1}{2} u_{+} + \frac{1}{2} u_{-} .

Applying the reciprocal eigenvalues of the inverse matrix therefore yields

y = \frac{1}{6} u_{+} - \frac{1}{2} u_{-} .

(- 1/3, 2/3)^{T}

S y = (1, 0)^{T}

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