Question Number 180923 by Shrinava last updated on 19/Nov/22
Answered by aleks041103 last updated on 20/Nov/22
$${Forbenius}:\:{A},{B},{C}\in{M}_{{n}} \left(\mathbb{F}\right) \\ $$$${rk}\left({abc}\right)+{rk}\left({b}\right)\geqslant{rk}\left({ab}\right)+{rk}\left({bc}\right) \\ $$$${b}={A} \\ $$$${a}={A}^{\mathrm{3}} \\ $$$${c}={A} \\ $$$$\Rightarrow{rk}\left({AAA}^{\mathrm{3}} \right)+{rk}\left({A}\right)\geqslant{rk}\left({A}^{\mathrm{3}} {A}\right)+{rk}\left({AA}\right) \\ $$$$\Rightarrow{rank}\left({A}^{\mathrm{5}} \right)+{rank}\left({A}\right)\geqslant{rank}\left({A}^{\mathrm{4}} \right)+{rank}\left({A}^{\mathrm{2}} \right) \\ $$
Commented by Shrinava last updated on 20/Nov/22
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor}\:\mathrm{cool} \\ $$