Show that $B^{-}A^{-}$ is a $g-inverse$ of $AB$ if and only if $A^{-}AB B^{-}$ is idempotent.
My work:
$AB(B^{-}A^{-})AB=AB$ Premultiplication and postmultiplication of both sides of the first of these two equalities by $A^{-}$, and $B^{-}$ respectively gives,
Forward: $ \ A^{-} AB(B^{-}A^{-})ABB^{-}=A^{-}ABB^{-}$
$ ( \ A^{-} ABB^{-})^2=A^{-}ABB^{-}$
For backward direction:
$A^{-} ABB^{-} A^{-} ABB^{-}=A^{-}ABB^{-}$
Multiplying left side and right side by B ,
$A A^{-} ABB^{-} A^{-} ABB^{-}B=AA^{-}ABB^{-}B$
After simplifying,
$A A^{-} ABB^{-} A^{-} ABB^{-}B=AA^{-}ABB^{-}B$
How to simplify the above line? I am kinda confused. Also I am not sure about wording of the left and right multiplication by A , and B. Thank you so much!