UPSC NDA II 2014 Mathematics Paper

AB = A
⇒ B × AB = B × A
⇒ (BA)B = B (∵ BA = B)
⇒ B × B = B
⇒ B2=B

Similarly BA = B
⇒A × BA = A × B
⇒(AB) A = A(∵ AB = A)
⇒ A × A = A
⇒A2=A

Similarly AB = A
⇒ (AB) (AB) =(AB) A
⇒(AB)2=A(BA)
⇒(AB)2=AB
Hence, all three statements are correct.