 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ia/paper-1/8a/a/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

M = A - λ I

M^{†} = A^{†} - \overline{λ} I

, so

M^{†} M M M^{†} = A^{†} A - λ A^{†} - \overline{λ} A + ∣ λ ∣^{2} I, = A A^{†} - \overline{λ} A - λ A^{†} + ∣ λ ∣^{2} I .

Since

A

is a normal matrix, the two products coincide. Therefore $A - λ I$ is normal for every complex $λ$ .

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