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 $Av$:
$$Av =
\begin{bmatrix} 1 & 3 \\ 2 & 2 \end{bmatrix}
\begin{bmatrix} 1 \\ 1 \end{bmatrix}
=
\begin{bmatrix} (1)(1) + (3)(1) \\ (2)(1) + (2)(1) \end{bmatrix} =
\begin{bmatrix} 4 \\ 4 \end{bmatrix}
$$
3. Calculate $\lambda v$:
$$\lambda v = 4 \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 4 \end{bmatrix}$$
4. Since they satisfy equation 7.1.1, $\lambda$ and $v$ are an eigenvalue/eigenvector pair for this matrix $A$.
