# How to prove that $c=(Id-B)A^{-1}b \hspace{1cm}$ if $x^{k+1}=Bx^{k}+c$ converges to the solution of $Ax=b$

Prove that if $$x^{k+1}=Bx^{k}+c$$, converge to the solution of $$Ax=b$$ then $$c=(Id-B)A^{-1}b$$

Taking limit on $$x_{n+1}=Bx_n+c$$ as $$n\to \infty$$ $$x=Bx+c\Rightarrow c=(I-B)x=(I-B)A^{-1}b.$$
• Where is the $d$ of $(Id-B)A^{-1}b$? – JuanMuñoz Aug 23 at 14:33
• There's no $d$, like $d=1$. Anything in the solution unclear? Maybe it has a mistake somewhere? Thanks. – Alexey Burdin Aug 23 at 14:39