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Question Number 44708 by rahul 19 last updated on 03/Oct/18

$$LetA,Bbetwon\times nmatricessuch$$$$thatA+B=ABthenprove:$$$$AB=BA?$$

Answered by arcana last updated on 03/Oct/18

$$\mathrm{B}=\mathrm{A}(\mathrm{B}-\mathrm{I})$$$$\mathrm{AB}={\mathrm{A}}^2(\mathrm{B}-\mathrm{I})=\mathrm{A}\left(\mathrm{AB}\right)-{\mathrm{A}}^2$$$$\mathrm{con}\mathrm{lo}\mathrm{nterior}\mathrm{llegamos}\mathrm{a}\mathrm{AB}(\mathrm{A}-\mathrm{I})={\mathrm{A}}^2$$$$\mathrm{reemplazando}\mathrm{es}\mathrm{AB}=\mathrm{AB}(\mathrm{A}-\mathrm{I})(\mathrm{B}-\mathrm{I})$$$$\mathrm{luego}\mathrm{necesariamente}\mathrm{se}\mathrm{tiene}$$$$\Rightarrow (\mathrm{B}-\mathrm{I})(\mathrm{A}-\mathrm{I})=\mathrm{I}$$$${(\mathrm{A}-\mathrm{I})}^{-1}=\mathrm{B}-\mathrm{I}$$$$\mathrm{la}\mathrm{matriz}\mathrm{inversa}\mathrm{a}\mathrm{izquierda}\mathrm{coincide}$$$$\mathrm{con}\mathrm{la}\mathrm{matriz}\mathrm{inversa}\mathrm{a}\mathrm{derecha}$$$$(\mathrm{A}-\mathrm{I})(\mathrm{B}-\mathrm{I})=\mathrm{I}$$$$\mathrm{asi},\mathrm{BA}-\mathrm{A}-\mathrm{B}=0_{\mathrm{n}\times \mathrm{n}}$$$$\Rightarrow \mathrm{BA}=\mathrm{A}+\mathrm{B}=\mathrm{AB}$$

Commented by rahul 19 last updated on 04/Oct/18

$$Thankssir!$$$$ButplswriteinEnglishfromnow$$$$onwards,ifounditverydifficult$$$$tounderstand.$$

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