Let $A, X_0 \in \mathbb{R}^{n\times n}$ and $det(A) \ne 0$.
Define the following reccurence: $$X_{k+1} = X_k + X_k(I-AX_k)$$
Prove $$\lim_{x \rightarrow \infty} X_k = A^{-1} \iff \rho(I-AX_k) < 1$$ where $\rho(B)$ means the biggest eigenvalue (in abs value) of the matrix $B$.
Any hints on how to prove it?