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Question Number 180923 by Shrinava last updated on 19/Nov/22

Answered by aleks041103 last updated on 20/Nov/22

$$Forbenius:A,B,C\in M_n\left(\mathbb{F}\right)$$$$rk\left(abc\right)+rk\left(b\right)⩾rk\left(ab\right)+rk\left(bc\right)$$$$b=A$$$$a=A^3$$$$c=A$$$$\Rightarrow rk\left({AAA}^3\right)+rk\left(A\right)⩾rk\left(A^{3}A\right)+rk\left(AA\right)$$$$\Rightarrow rank\left(A^5\right)+rank\left(A\right)⩾rank\left(A^4\right)+rank\left(A^2\right)$$

Commented by Shrinava last updated on 20/Nov/22

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professor}\mathrm{cool}$$

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