Non singular matrices proof

Question

The following problem is from an unknown book, if you know how the book is called, please tell me.

Let A be a non-singular matrix, $x$ and $y \in C^n$ . Show that:

Thank you, I don't know even how to start with this problem.

Answer 1

This is a special case of the Woodbury matrix identity; you can verify the first line directly by using the definition of an inverse. \begin{align} (A^{-1} + xy^*)\left(A - \frac{Axy^* A}{1+y^* Ax}\right) &= I - \frac{xy^* A}{1+y^* Ax} + xy^*A - \frac{xy^* Axy^* A}{1+y^* Ax}\\ &= I + xy^* A - \frac{x(1 + y^* A x)y^* A}{1+y^* Ax} \\ &= I + xy^* A - xy^* A\\ &= I. \end{align}