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

sagnuhh 2020-10-19 Answered

Let MR10×10,s.t.M2020=0. Prove M10=0

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Expert Answer

Ayesha Gomez
Answered 2020-10-20 Author has 104 answers

Let A10×10 matrix. Given that A2020=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)=xk 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<10 then A10=A10kAk=A10k×0=0.
In Any case we have A10=0.

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