Linear Algebra Proof using nonsingularity

Question

In my Linear Algebra course I have come across multiple questions of the sort which I cannot seem to answer.

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 and

$(A − BC)^{-1} = A^{-1} + A^{-1}B(I − CA^{-1}B)^{-1}CA^{-1}$.

Edit: Any places where I can find rules and tricks on solving these kinds of matrix equations would be greatly appreciated.

Answer 1

Just by multiplying $A-BC$ and $A^{-1} + A^{-1}B(I - CA^{-1}B)^{-1}CA^{-1}$ you get the desired result, I strongly suggest you to try it yourself before looking at the following.

\begin{align*} &(A-BC)(A^{-1} + A^{-1}B(I - CA^{-1}B)^{-1}CA^{-1})\\ =&I-BCA^{-1}+B(I - CA^{-1}B)^{-1}CA^{-1}-BCA^{-1}B(I - CA^{-1}B)^{-1}CA^{-1}\\ =&I-BCA^{-1}+B(I-CA^{-1}B)(I - CA^{-1}B)^{-1}CA^{-1}\\ =&I-BCA^{-1}+BCA^{-1}\\ =&I \end{align*}