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Involutory Matrix

A square matr...

Question

A square matrix $A$ is said to be an idempotent matrix if $A^{2} = A$ . If $A$ and $B$ are idempotent matrices and $A B = B A$ , then which of the following is an idempotent matrix

A + B

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A B

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A - B

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B - A

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Solution

The correct option is B $A B$
$A a n d B a r e i d e m p o t e n t m a t r i c e s a n d A B = B A i . e A^{2} = A a n d B^{2} = B {(A B)}^{2} = (A B) (A B) = A (B A) B = A (A B) B = (A A) (B B) \Rightarrow {(A B)}^{2} = A B H e n c e, A B i s a n i d e m p o t e n t m a t r i x .$

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