Finding an Eigenvector of $3\times 3$ matrix

Question

I have a question $\lambda=4$ find an Eigenvector of a given $3\times 3$ matrix.

$ A = \left[ {\begin{array}{cc} 1 & 2 & 1 \\ 6 & 1& 0 \\ -1 & -2 & -1 \end{array} } \right] $

I know the answer which is

$ A = \left[ {\begin{array}{cc} 1 \\ -2 \\ -1 \end{array} } \right] $

But I don't know how to solve it?

The way I tried to do is with RREF but I am not sure how to present it. So can someone help me please

RREF $ A = \left[ {\begin{array}{cc} 1 & -2 & -1 \\ 0 & 13 & 6 \\ 0 & 0 & 0 \end{array} } \right] $

Answer 1

Hint

Pick $x=(x_1,x_2,x_3)^T$ and solve the system of three equations: $$Ax=4x$$

Answer 2

you only need to solve $\left( \begin{array}{ c c } 1 & 2 & 1 \\ 6 & 1 & 0 \\ -1 & -2 & -1 \end{array} \right) * \left( \begin{array}{ c c c } u \\ v \\ w \end{array} \right)= \lambda *\left( \begin{array}{ c c } u \\ v \\ w \end{array} \right)$