Determine the eigenspace and its dimension for the following matrix

Question

The Matrix $\begin{bmatrix} -2 & -1 & -1 \\ 0 & 0 & 2 \\ -1 & 0 & -2 \end{bmatrix} \in M(3,3,\mathbb{R})$ is given with eigenvalue $\lambda_1 = -2$

I am required to

i) "determine the eigenspace" for said value

ii) "determine the dimension of the eigenspace, i.e the space of all eigenvectors for $\lambda_1 = -2$

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EDIT: One of the assumptions I made in the solution below is that $0 \cdot x_1 = 0$ means that I can ignore it in the eigenspace.

Progress so far:

substituting the value of $\lambda_1 = -2$ yields $\begin{bmatrix} 0 & -1 & -1 \\ 0 & 2 & 2 \\ -1 & 0 & 0 \end{bmatrix} \overset{\mbox{RREF}}{\longrightarrow} $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix}

Solving $(A - \lambda I)\vec{x} = \vec{0}$ and rearranging yields: $x_1 = 0$ and $x_2 = -x_3$.

Am I correct in assuming that the eigenspace is just $\{ \begin{bmatrix} 0 \\ -x \\ x\end{bmatrix} : x \in \mathbb{R}\}$?

If so, is it correct to conclude that for part II, I can simply count the vectors produced in part I (i.e 1) and the question has been fully completed?

Thanks in advance.

Answer 1

Note that $$ \begin{bmatrix} 0 \\ -x \\ x\end{bmatrix} = x \begin{bmatrix} 0 \\ -1 \\ 1\end{bmatrix} $$

Thus your eigenspace associated to $\lambda_1=-2$ is a one dimensional subspace generated by $ \begin{bmatrix} 0 \\ -1 \\ 1\end{bmatrix} $