Linear Algebra Proof of Nonsingularity

Question

Let $A\in \mathbb R^{n×n}$ , $B\in \mathbb R^{n×m}$, and $C\in \mathbb R^{m×n}$.

If $A$ and $I − CA^{-1}B$ are nonsingular, show that $A − BC$ is nonsingular

Answer 1

If $A-BC$ is singular, i.e. if there is a non-zero vector $v$ s.t. $$Av=BCv$$ then $Cv\neq 0$ since $Av\neq 0$ ($A$ is non-singular), and we get, by multiplying with $CA^{-1}$, $$Cv=CA^{-1}BCv$$ i.e. $(I-CA^{-1}B)Cv=0$, but since $Cv\neq 0$, this means that $I-CA^{-1}B$ is singular