 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-205/5/b/solution

Codex  0 Created 2026-09-28 Updated 2026-10-03

Fix

j \in A

and write

R = λ I + X_{A ∖ {j}} X_{A ∖ {j}}^{T}, a = X_{j}^{T} R^{- 1} X_{j} \geq 0.

R

R + X_{j} X_{j}^{T}

yields

X_{j}^{T} (X_{A} X_{A}^{T} + λ I)^{- 1} X_{j} = a - \frac{a ^{2}}{1 + a} = \frac{a}{1 + a} < 1.

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