Answer
Since they satisfy equation 7.1.1, $\lambda$ and $v$ are an eigenvalue/eigenvector pair for this matrix $A$.
Work Step by Step
1. According to equation 7.1.1:
$$Av = \lambda v$$
2. Calculate $Av$:
$$Av =
\begin{bmatrix} 1 & -2 & -6 \\ -2 & 2 & -5 \\ 2 & 1 & 8 \end{bmatrix}
\begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}
=
\begin{bmatrix} (1)(2) + (-2)(1) + (-6)(-1) \\ (-2)(2) + (2)(1) + (-5)(-1) \\ (2)(2) + (1)(1) + (8)(-1) \end{bmatrix} =
\begin{bmatrix} 6 \\ 3 \\ -3\end{bmatrix}
$$
3. Calculate $\lambda v$:
$$\lambda v = 3 \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 6 \\ 3
\\ -3 \end{bmatrix}$$
4. Since they satisfy equation 7.1.1, $\lambda$ and $v$ are an eigenvalue/eigenvector pair for this matrix $A$.
