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Question

If A and B are two matrices such that AB=A and BA=B, then


A
(A3B3)5=(A5B5)3
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B
(A4B4)3=A12B12
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C
(AB) is idempotent
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D
(A10B10) is nilpotent
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Solution

The correct options are
A (A3B3)5=(A5B5)3
D (A10B10) is nilpotent
AB=AABA=A2A2=A
Similarly B2=B
A and B are idempotent
Also (AB)2=(AB)(AB)
=A2ABBA+B2=AAB+B=0
(AB) is nilpotent

Mathematics

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