 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/3/solution

(1, 1)^{T}

and

(1, - 1)^{T}

are eigenvectors: multiplication adds or subtracts the two entries in a row. Their eigenvalues are therefore

λ_{+} = e^{ϕ} + e^{- ϕ} = 2 cosh ϕ, λ_{-} = e^{ϕ} - e^{- ϕ} = 2 sinh ϕ .

The second eigenvalue may be positive, zero or negative as the real parameter changes.

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