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Let-A-B-be-two-n-n-matrices-such-that-A-B-AB-then-prove-AB-BA-

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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 ?
LetA,Bbetwon×nmatricessuchthatA+B=ABthenprove: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
B=A(B−I)AB=A2(B−I)=A(AB)−A2conlonteriorllegamosaAB(A−I)=A2reemplazandoesAB=AB(A−I)(B−I)luegonecesariamentesetiene⇒(B−I)(A−I)=I(A−I)−1=B−Ilamatrizinversaaizquierdacoincideconlamatrizinversaaderecha(A−I)(B−I)=Iasi,BA−A−B=0n×n⇒BA=A+B=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.
Thankssir!ButplswriteinEnglishfromnowonwards,ifounditverydifficulttounderstand.

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