$(AB)^-=B^-A^{-1}$ holds when $A$ is nonsingular and $B$ is singular?

Question

Suppose that $A$ is a nonsingular and $B$ is a singular $n\times n$ matrix. $B^-$ is a generalized inverse of $B$. The following statement is valid?

$(AB)^-=B^-A^{-1}$

Answer 1

A generalized inverse $(AB)^-$ must satisfy the equation $$ AB(AB)^-AB=AB. $$

Since $A$ is nonsingular, it is enough to check $$ B(B^-A^{-1})AB=B. $$

But this holds since cancelling $A^{-1}$ with $A$, this amounts to $BB^-B=B$, which is the defining property of a generalized inverse.