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Question Number 206609 by universe last updated on 20/Apr/24

$$\mathrm{let}\mathrm{matrix}{\mathrm{A}}_{\mathrm{n}\times \mathrm{n}}\mathrm{and}{\mathrm{B}}_{\mathrm{n}\times \mathrm{n}}\mathrm{satisfying}$$$${\mathrm{A}}^2=\mathrm{A}\mathrm{\&}{\mathrm{B}}^2=\mathrm{B}\mathrm{then}\mathrm{prove}$$$$\rho (\mathrm{A}-\mathrm{B})=\rho (\mathrm{A}-\mathrm{AB})+\rho (\mathrm{B}-\mathrm{AB})$$$$\mathrm{here}\rho =\mathrm{rank}$$

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