 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/iii/solution

Since every eigenvalue is nonzero,

S

is an invertible matrix. Multiplying

S u_{i} = λ_{i} u_{i}

S^{- 1} u_{i} = λ_{i}^{- 1} u_{i} .

Hence the eigenvectors are unchanged and the corresponding eigenvalues are

1/ λ_{i}

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