 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-1/1/c/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Let

S_{m} = I + D + \dots + D^{m}

(I - D) S_{m} = I - D^{m + 1}

. By the assumed convergence, its terms

D^{m + 1}

tend to zero, so the limit

S

satisfies

(I - D) S = I

. Consequently

S = (I - D)^{- 1}

. This is the finite-dimensional Neumann series argument; invertibility by itself would not imply convergence.

For the stated nilpotent matrix,

D^{3} = 0

and only the first three terms remain:

(I - D)^{- 1} = I + D + D^{2} = 100110111 .

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