 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/ii/solution

Insert the eigenvalues and eigenvectors into the inverse spectral decomposition. The first two projections contribute

\frac{1}{2} e_{1} e_{1}^{T} + \frac{1}{4} e_{2} e_{2}^{T} = 3/8 1/8 0 1/8 3/8 0 000 .

Adding the third projection gives

A^{- 1} = 3/8 1/8 0 1/8 3/8 0 001 .

A

gives the identity; in particular the upper block satisfies

(3 - 1 - 1 3) (3113) = 8 I

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