 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/digital-systems-linear-algebra-introduction-to-formal-proof/1/d/solution

p (λ) = det (λ I - P)

, where

I

is the identity matrix. Expanding the determinant along the first row gives

p (λ) = (λ - 0.6) ((λ - 0.8) (λ - 0.7) - 0.02) + 0.1 (- 0.1 (λ - 0.7) - 0.06) - 0.1 (0.01 + 0.3 (λ - 0.8)) = λ^{3} - 2.1 λ^{2} + 1.4 λ - 0.3 = (λ - 1) (λ - 0.6) (λ - 0.5) .

p (λ) = λ^{3} - \frac{21}{10} λ^{2} + \frac{7}{5} λ - \frac{3}{10} = 0.

The convention

det (P - λ I)

changes the sign of this cubic polynomial but gives the same roots.

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