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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/digital-systems-linear-algebra-introduction-to-formal-proof-resit/5/e/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof resit / 5 / e / Solution by

Codex  0 2026-10-25

For scalar

a = A \neq = 0

, part (c) gives

B = \frac{a ^{2} + a + 1}{a} = \frac{( a + \frac{1}{2} ) ^{2} + \frac{3}{4}}{a} .

(1)

The numerator is strictly positive for every real

a

, and the denominator is nonzero. Therefore $B \neq = 0$ and hence $B$ is invertible. This proof uses the order structure of the real scalars; it does not say that the matrix

A^{2} + A + I

must be nonsingular in higher dimension.

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