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Aand $\mathrm{B}$ are two matrices such that $\mathrm{AB}=\overline{\mathrm{A}}$ and $\mathrm{BA}=\mathrm{B}$ then what is $\mathrm{B}^{2}$ equal to?
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The correct answer is:
$\mathrm{B}$
Given $\mathrm{AB}=\mathrm{A}$ and $\mathrm{BA}=\mathrm{B}$
Consider $\mathrm{B}=\mathrm{BA}=\mathrm{B}(\mathrm{AB})(\because \mathrm{AB}=\mathrm{A})$
$=(\mathrm{BA}) \cdot \mathrm{B}=\mathrm{B} \cdot \mathrm{B}=\mathrm{B}^{2}$
Hence, $\mathrm{B}^{2}=\mathrm{B}$
Consider $\mathrm{B}=\mathrm{BA}=\mathrm{B}(\mathrm{AB})(\because \mathrm{AB}=\mathrm{A})$
$=(\mathrm{BA}) \cdot \mathrm{B}=\mathrm{B} \cdot \mathrm{B}=\mathrm{B}^{2}$
Hence, $\mathrm{B}^{2}=\mathrm{B}$
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