Question

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

Matrices

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

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 $$A^{10}=0$$.