 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/13/a/iii/solution

det (A - λ I) = (5 - λ)^{2} - 9 = (λ - 2) (λ - 8) .

For

λ = 2

x = y

; for

λ = 8

, it reduces to

x = - y

. Choose the following unit vectors consistently for the remaining parts:

λ_{1} = 2, x_{1} = \frac{1}{2} (11); λ_{2} = 8, x_{2} = \frac{1}{2} (1 - 1) .

Each eigenspace consists of all scalar multiples of its displayed vector, including zero; an eigenvector itself must be nonzero.

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