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Question Number 147312 by mathdanisur last updated on 19/Jul/21

Answered by Olaf_Thorendsen last updated on 19/Jul/21

$$\mathrm{A}+\mathrm{B}=\mathrm{AB}$$$$\iff \mathrm{AB}-\mathrm{A}-\mathrm{B}=\stackrel{\sim }{0}$$$$\iff \mathrm{AB}-\mathrm{A}-\mathrm{B}+{\mathrm{I}}_n={\mathrm{I}}_n$$$$\iff (\mathrm{A}-{\mathrm{I}}_n)(\mathrm{B}-{\mathrm{I}}_n)={\mathrm{I}}_n$$$$$$$$\mathrm{A}+\mathrm{B}=\mathrm{AB}\iff {(\mathrm{A}-{\mathrm{I}}_n)}^{-1}\mathrm{exists}.$$$$\mathrm{In}\mathrm{this}\mathrm{case}{(\mathrm{A}-{\mathrm{I}}_n)}^{-1}=\mathrm{B}-{\mathrm{I}}_n$$

Commented by mathdanisur last updated on 20/Jul/21

$$thanksSer,butA=B=0,A=B=21$$

Commented by mathdanisur last updated on 20/Jul/21

$$sorryA=B=2IandA=B=0$$

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