 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-i/3/a/iii/solution

λ

is an eigenvalue of

A

if and only if there is a nonzero vector

v

with

A v = λ v

. By the preceding kernel criterion,

A v = λ v for some v \neq = 0 ⟺ (λ I_{n} - A) v = 0 for some v \neq = 0 ⟺ det (λ I_{n} - A) = 0.

The last condition is

χ_{A} (λ) = 0

, exactly the definition of a polynomial root. For a real matrix, this proves the assertion for real eigenvalues; the same proof over the complex numbers gives the usual complex version.

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