Suppose I am trying to solve for an eigenvector provided my matrix $$A=\left( \begin{array}{cc} 2 & 7 \\ 7 & 2 \\ \end{array} \right)$$
eigenvalues are $9$ and $-5$
How can I get an eigenvector when I set lambda to $-5$?
$$ \begin{array}{cc|c} -3 & 7&0 \\ 7 & -3&0 \\ \end{array} $$
$$\left( \begin{array}{cc} 2 & 7 \\ 7 & 2 \\ \end{array} \right)-(-5)I=(0,0)$$ $$ \left( \begin{array}{cc} 7 & 7 \\ 7 & 7 \\ \end{array} \right)\left( \begin{array}{c} x \\ y \\ \end{array} \right)=\left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right)$$ $7x+7y=0\to y=-x$ so eigenvector is $(1,-1)$
In a similar way for the other eigenvalue you get $(1,1)$