Ques. -28

Algebra Level 3

If A A and B B are square matrices such that A 2 = A , B 2 = B A^2=A, B^2=B and A , B A,B commute, then:

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( A B ) 2 = A B (AB)^2=AB ( A B ) 2 = I (AB)^2=I ( A B ) 2 = 0 (AB)^2=0 ( A B ) 2 = − A B (AB)^2=-AB

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Francis Kong
May 19, 2018

Short Answer:

( A B ) 2 = ( A B ) ( A B ) = A B A B = A A B B = A 2 B 2 = A B (AB)^2 = (AB)(AB) = ABAB = AABB = A^2B^2=AB

Long Answer:

( A B ) 2 = ( A B ) ( A B ) = A B A B (AB)^2 = (AB)(AB) = ABAB (Expanding)

A B A B = A A B B ABAB=AABB Given: A A , B B Commute means A B = B A AB=BA

A A B B = A 2 B 2 AABB = A^2B^2 (Grouping like terms)

A 2 B 2 = A B A^2B^2=AB Given: A 2 = A A^2=A and B 2 = B B^2=B

Hence, ( A B ) 2 = A B \boxed{(AB)^2=AB}

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