Given $\lambda$ is -2 and after calculating $A-\lambda I$, $$\left(\begin{array}{rrr} 3 & 3 & 3\\ -3 & -3 & -3\\ 3 & 3 & 3 \end{array}\right).$$
The reduced row echelon form is: $$\left(\begin{array}{ccc} 1 & 1 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right).$$
The solution is actually: $$\left(\begin{array}{aaa} -1\\ 1\\ 0 \end{array}\right).$$ and
$$\left(\begin{array}{bbb} -1\\ 0\\ 1 \end{array}\right).$$
Can someone explain how the book arrived at this?