 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-2/4/solution

(1 - λ)^{2} - 4 = (λ - 3) (λ + 1)

. For

λ = 3

the components are equal; for

λ = - 1

they are opposite. After normalization, the eigenvalues and associated eigenvectors are

λ_{1} = 3, v_{1} = \frac{( 1 , 1 )}{2}; λ_{2} = - 1, v_{2} = \frac{( 1 , - 1 )}{2} .

An overall sign change of either normalized eigenvector is equivalent.

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