A) \[{{({{A}^{5}}-{{B}^{5}})}^{3}}=A-B\]
B) \[{{({{A}^{5}}-{{B}^{5}})}^{3}}={{A}^{3}}-{{B}^{3}}\]
C) \[A-B\]is idempotent
D) \[A-B\]is nilpotent
Correct Answer: D
Solution :
[d] Since AB=B and BA=A, so \[BAB={{B}^{2}}\] Or \[(BA)B={{B}^{2}}\] Or \[AB={{B}^{2}}\] Or \[B={{B}^{2}}\] Hence, B is idempotent and similarly A. \[{{(A-B)}^{2}}={{A}^{2}}-AB-BA+{{B}^{2}}\] \[=A-B-A+B=0\] Therefore, A-B is nilpotent.
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