It is known that A is a matrix over the number field k, and a k = 0, so | a | k = 0, so |A|=0.

Therefore | 0i-a | =|-a| = (- 1) n | a | = 0.

So 0 is the eigenvalue of a.

Let λ be an eigenvalue of a, then α∈K(n) Aα=λα≠0 exists, so α = λ α. Multiply both sides to the left by a, a 2 * α = a (λ α) = λ (α α) = λ 2 * α. Continuing this process, we can get a kα = λ kα. Since a k = 0, λ k * α = 0. Because α≠0, therefore,

λ^k=0。 So λ=0

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This is a topic about matrix eigenvalue. I suggest that you take a look at the chapter "Similarity and Bias of Matrices" on page 324 of Advanced Algebra Learning Guide published by Tsinghua University Publishing House.

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