Question

If $AB=A$ and $BA=B$ and $B^{\prime }{A}^{\prime }=A^{\prime }$ and $A^{\prime }{B}^{\prime }=B^{\prime }$ then prove that $A^{\prime }$ and $B^{\prime }$ are idempotent.

Answer 1

$B^{\prime }{A}^{\prime }=(AB{)}^{\prime }=A^{\prime }$ and $A^{\prime }{B}^{\prime }=(BA{)}^{\prime }=B^{\prime }$
Now $A^{\prime }$ is idempotent if ${A^{\prime }}^2=A^{\prime }$
Then ${A^{\prime }}^2=A^{\prime }{A}^{\prime }=A^{\prime }\left(B^{\prime }{A}^{\prime }\right)=\left(A^{\prime }{B}^{\prime }\right)A^{\prime }=B^{\prime }{A}^{\prime }=A^{\prime }$
$\therefore$ $A^{\prime }$ is idempotent. Similarly $B^{\prime }$ is idempotent.