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

Let A,B be two n×n matrices such that A+B=AB then prove : AB=BA ?

$${Let}\:{A},{B}\:{be}\:{two}\:{n}×{n}\:{matrices}\:{such} \\ $$$${that}\:{A}+{B}={AB}\:{then}\:{prove}\:: \\ $$$${AB}={BA}\:? \\ $$

Answered by arcana last updated on 03/Oct/18

B=A(B−I) AB=A^2 (B−I)=A(AB)−A^2 con lo nterior llegamos a AB(A−I)=A^2 reemplazando es AB=AB(A−I)(B−I) luego necesariamente se tiene ⇒(B−I)(A−I)=I (A−I)^(−1) =B−I la matriz inversa a izquierda coincide con la matriz inversa a derecha (A−I)(B−I)=I asi, BA−A−B=0_(n×n) ⇒BA=A+B=AB

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

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

Thanks sir! But pls write in English from now onwards , i found it very difficult to understand.

$${Thanks}\:{sir}! \\ $$$${But}\:{pls}\:{write}\:{in}\:{English}\:{from}\:{now} \\ $$$${onwards}\:,\:{i}\:{found}\:{it}\:{very}\:{difficult} \\ $$$${to}\:{understand}. \\ $$

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