How to compute Eigenvectors from Eigenvalues?

How can I use the formula below to find the first eigenvector for the matrix $$A$$?

The matrix:
$$A= \begin{pmatrix} 2 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -1 & 2 \\ \end{pmatrix}$$ with eigenvalues: $$-1, 2$$ and $$3$$

The formula: $$|v_{i,j}|^2\prod_{k=1;k\neq i}^n(\lambda_i(A)-\lambda_k(A))=\prod_{k=1}^{n-1}(\lambda_i(A)-\lambda_k(M_j))$$

Source of the formula: https://arxiv.org/abs/1908.03795

Also, I try to found the eigenvalues of the minor matrix by deleting the first column and row of the matrix $$B$$ and I get complex ones, should I continue searching the elements with the formula despite that? $$B= \begin{pmatrix} 1 & 3 & 0 \\ 1 & 0 & 1 \\ 0 & -1 & 1 \\ \end{pmatrix}$$

And thank you.