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Question 7.5.9: Consider an n × n matrix A with complex eigenvalues λ1, λ2,.......

Consider an n × n matrix A with complex eigenvalues λ_1, λ_2,…,λ_n, listed with their algebraic multiplicities. What is the relationship between the λ_i and the determinant of A? Compare with Theorem 7.2.8.

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f_A(λ) = \det(A − λI_n) = (λ_1 − λ)(λ_2 − λ)···(λ_n − λ)

f_A(0) = \det A = λ_1λ_2 ··· λ_n

so that

\det A = λ_1λ_2 ··· λ_n

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