Question

Let M in R^{10×10}, s.t. M^{2020} =0. Prove M^{10} = 0

Matrices
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asked 2020-10-19

Let \(M \in R^{10×10}, s.t. M^{2020} =0\). Prove \(M^{10} = 0\)

Answers (1)

2020-10-20

Let \(A10×10\) matrix. Given that \(A^{2020}=0\). Let m(x) be the minimal polynomial of A. By Cayley-Hamilton, the degree of mm is atmost 10. By the propert of minimal polynomial m(x) devides x2020. Then the possibilty of are \(m(x)=x^{k}\) for some k=1,2,⋯ ,10.
Case 1: If $k=10, $ and since A satisfies its minial polynomial we have \(A^{10}=0\).
Case 2: If k<10 then \(A^{10}=A^{10}-kA^{k}=A10-k\times0=0.\)
In Any case we have \(A^{10}=0\).

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