Is this an acceptable way to find an eigenvalue?

Question

I have a matrix M where $$ M = \begin{pmatrix} -2 & 2 & 2 \\ 2 & 1 & 2 \\ -3 & -6 & -7 \\ \end{pmatrix} $$ and it has an eigenvector of $\begin{pmatrix} 0 \\ 1 \\ -1 \\ \end{pmatrix}$

I am trying to find the corresponding eigenvalue but I am not sure if it is the correct way to do it. My working is shown below: $$\begin{align} (M - \lambda I)\cdot x & = 0 \\ \begin{pmatrix} -2 - \lambda & 2 & 2 \\ 2 & 1 - \lambda & 2 \\ -3 & -6 & -7 - \lambda\\ \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 1 \\ -1 \\ \end{pmatrix} & = 0 \\ \begin{pmatrix} 0 \\ 1 - \lambda -2 \\ -6 + 7 + \lambda \\ \end{pmatrix} & = 0 \\ \end{align}$$ Therefore, $1- \lambda - 2 = 0$ or $-6 + 7 + \lambda = 0$ so $\lambda = -1$

Please correct me if my presentation isn't right.

Answer 1

Why not use the definition $M\cdot V=\lambda V$

In our case

$$M\cdot V=\begin{bmatrix} 0\\-1\\1\end{bmatrix}=-V$$

So the eigenvalue is $-1$