Given $A, x, b$ and a linear system $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is full rank. Denote $A_{\backslash i}$ as a $(n-1)\times(n-1)$ matrix where $A$ removes its $i^{th}$ column and row, similarly $b_{\backslash i}$. What's the most efficient way to find a $y$ for each $i\in[n]$ s.t $A_{\backslash i}y=b_{\backslash i}$? (So in total these are $n$ linear systems, for different $i$ I want a different solution $y$).

I know one way is to use Sherman-Morrison formula, but is there any way that doesn't involve calculating $A^{-1}$? That is, suppose I have $x$ calculated, can $y$ be derived just using $A, x, b$?

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