Why doesn't $AA^-A=A \Rightarrow A^-AA^-=A^-$?

Question

The condition for being a generalized inverse matrix is $AA^-A=A$.

There's another condition $A^-AA^-=A^-$, and when this also holds, $A^-$ is called a reflexive inverse.

But when does it happen that only the first of these conditions holds? Doesn't the first condition imply the second?

Answer 1

Consider $A=\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}$ and $A^{-}=I$. This satisfies the first condition but not the second.