 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-1/4/i/solution

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

Decompose

R^{n} = span {a} \oplus a^{⊥}

a a^{T}

∥ a ∥^{2}

along

a

and as zero on its orthogonal complement. Thus

λ_{∥} (A^{\pm}) = 1 \pm ∥ a ∥^{2}, v_{∥} \in span {a} ∖ {0}; λ_{⊥} = 1, v_{⊥} \in a^{⊥} ∖ {0} .

n - 1

and is absent when

n = 1

. Consequently

A^{+}

is always an invertible matrix, while

A^{-} is invertible if and only if ∥ a ∥ \neq = 1.

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