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

Question
Matrices
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 A10=0.
Case 2: If k\(A^{10}=A^{10}-kA^{k}=A10-k\times0=0.\)
In Any case we have A^{10}=0.
0

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