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Question Number 98369 by nesreen last updated on 13/Jun/20

$$\mathrm{If}A,B\mathrm{are}\mathrm{two}\mathrm{square}\mathrm{matrices}\mathrm{such}$$$$\mathrm{that}AB=A\mathrm{and}BA=B,\mathrm{then}$$

Answered by bobhans last updated on 13/Jun/20

$${\mathrm{A}}^{-1}\mathrm{AB}={\mathrm{A}}^{-1}\mathrm{A}\Rightarrow \mathrm{B}=\mathrm{I}$$$${\mathrm{B}}^{-1}\mathrm{BA}={\mathrm{B}}^{-1}\mathrm{B}\Rightarrow \mathrm{A}=\mathrm{I}$$$$\mathrm{then}\mathrm{A}=\mathrm{B}=\mathrm{I}$$

Answered by Rio Michael last updated on 13/Jun/20

$$\mathrm{Also}AB=A\mathrm{and}BA=B$$$$\Rightarrow A\left(BA\right)=A$$$$\Rightarrow \left(AB\right)A=A$$$$\Rightarrow \left(A\right)\left(A\right)=A$$$$\Rightarrow A^2=A\mathrm{similarly},B^2=B$$

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