 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-1/6c/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Write

a = (a_{1}, a_{2}, a_{3}, a_{4})^{T}

a

sends

x ⟼ x - 2 \frac{a ^{T} x}{a ^{T} a} a,

M = I_{4} - \frac{2 a a ^{T}}{a ^{T} a} .

It acts as

- 1

span {a}

and as

1

on the three-dimensional hyperplane

a^{⊥}

. Therefore

det M = - 1 .

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