I'm currently trying to solve the following statement regarding generalized inverse matrix $A^{-}$;
If $A^{-}$ is reflexive, then $(A^{-}A)' = A^{-}A$ and $(AA^{-})' = AA^{-}$
To start with, $A^{-}$ is reflexive, so $A$ is a generalize inverse of $A^{-}$. That is, $$A^{-} =A^{-}AA^{-}.$$
Then, I have to show $(A^{-}A)' =A'(A^{-})' = A^{-}A$, which seems unlcear. Simply substituting $A^{-}$ with $A^{-}AA^{-}$ does not give any useful information.
Any hints or other different way with respect to this problem would be grateful. Thank you.
As it is formulated, the statement is false.
Consider for instance: $$ A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix},\hspace{3mm} G = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}. $$ We then have $G = GAG$ (and, analogously, $A = AGA$). However: $$ GA = \begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix} \neq (GA)'. $$