 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/14/b/iii/solution

Summing the diagonal entries gives

tr (A^{- 1}) = \frac{3}{8} + \frac{3}{8} + 1 = \frac{7}{4} .

The eigenvalues of the inverse are the reciprocals

1/2, 1/4, 1

of those of

A

, whose sum is also

7/4

. This is precisely what the trace identity predicts.

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