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If $A$ and $B$ are matrices and $B=A B A^{-1}$ then the value of $(A+B)(A-B)$ is
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Verified Answer
The correct answer is:
$\mathrm{A}^{2}-\mathrm{B}^{2}$
$\mathrm{B}=\mathrm{ABA}^{-1}$ (Given)
But $B=B A A^{-A}$
$\therefore \mathrm{ABA}^{-1}=\mathrm{BAA}^{-1} \Rightarrow \mathrm{AB}=\mathrm{BA}$ $\mathrm{Now}(\mathrm{A}+\mathrm{B})(\mathrm{A}-\mathrm{B})=\mathrm{A}^{2}-\mathrm{AB}+\mathrm{BA}-\mathrm{B}^{2}$ $=\mathrm{A}^{2}-\mathrm{AB}+\mathrm{AB}-\mathrm{B}^{2} \quad[\because \mathrm{AB}=\mathrm{BA}]$ $=\mathrm{A}^{2}-\mathrm{B}^{2}$
But $B=B A A^{-A}$
$\therefore \mathrm{ABA}^{-1}=\mathrm{BAA}^{-1} \Rightarrow \mathrm{AB}=\mathrm{BA}$ $\mathrm{Now}(\mathrm{A}+\mathrm{B})(\mathrm{A}-\mathrm{B})=\mathrm{A}^{2}-\mathrm{AB}+\mathrm{BA}-\mathrm{B}^{2}$ $=\mathrm{A}^{2}-\mathrm{AB}+\mathrm{AB}-\mathrm{B}^{2} \quad[\because \mathrm{AB}=\mathrm{BA}]$ $=\mathrm{A}^{2}-\mathrm{B}^{2}$
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