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Question

# If AB = A and BA = B, where A and B are square matrices, then (a) B2 = B and A2 = A (b) B2 ≠ B and A2 = A (c) A2 ≠ A, B2 = B (d) A2 ≠ A, B2 ≠ B

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Solution

## (a) B2 = B and A2 = A $\mathrm{Here},\phantom{\rule{0ex}{0ex}}AB=A...\left(1\right)\phantom{\rule{0ex}{0ex}}BA=B...\left(2\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒ABA=AA\left[\mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}A\right]\phantom{\rule{0ex}{0ex}}BAB=BB\left[\mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}A\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒AB={A}^{2}\left[\mathrm{From}\mathrm{eq}.\left(2\right)\right]\phantom{\rule{0ex}{0ex}}BA={B}^{2}\left[\mathrm{From}\mathrm{eq}.\left(1\right)\right]\phantom{\rule{0ex}{0ex}}⇒A={A}^{2}\left[\mathrm{From}\mathrm{eq}.\left(1\right)\right]\phantom{\rule{0ex}{0ex}}B={B}^{2}\left[\mathrm{From}\mathrm{eq}.\left(2\right)\right]$

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