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 $Av$:
$$Av =
\begin{bmatrix} -7 & 2 \\ -9 & 4 \end{bmatrix}
\begin{bmatrix} 2 \\ 9 \end{bmatrix}
=
\begin{bmatrix} (-7)(2) + (2)(9) \\ (-9)(2) + (4)(9) \end{bmatrix} =
\begin{bmatrix} 4 \\ 18 \end{bmatrix}
$$
3. Calculate $\lambda v$:
$$\lambda v = 2 \begin{bmatrix} 2 \\ 9 \end{bmatrix} = \begin{bmatrix} 4 \\ 18 \end{bmatrix}$$
4. Since they satisfy equation 7.1.1, $\lambda$ and $v$ are an eigenvalue/eigenvector pair for this matrix $A$.
