 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/digital-systems-linear-algebra-introduction-to-formal-proof/6/d/solution

p (λ) = det (λ I - P)

, with

I

u = λ - 1/2

. Then

p (λ) = det u - 1/4 - 1/4 - 1/2 u 0 - 1/2 0 u = u^{3} - \frac{1}{8} u - \frac{1}{8} u = u (u^{2} - \frac{1}{4}) = λ (λ - 1/2) (λ - 1) .

p (λ) = λ^{3} - \frac{3}{2} λ^{2} + \frac{1}{2} λ = 0.

Using

det (P - λ I)

instead negates the cubic polynomial and leaves the eigenvalues unchanged.

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