Answer
See below
Work Step by Step
1. According to equation 7.1.1:
$$Av = \lambda v$$
2. Calculate $Av$:
$$Av =
\begin{bmatrix} 1 & 4 & 1 \\ 3 & 2 & 1 \\ 3 & 4 & -1 \end{bmatrix}
\begin{bmatrix} c_1+4c_2 \\ -3c_2 \\ -3c_1 \end{bmatrix} =\begin{bmatrix} -2c_1-8c_2 \\ 6c_2
\\ 6c_1 \end{bmatrix}
$$
3. Calculate $\lambda v$:
$$\lambda v = -2 \begin{bmatrix} c_1+4c_2 \\ -3c_2 \\ -3c_1 \end{bmatrix} = \begin{bmatrix} -2c_1-8c_2 \\ 6c_2
\\ 6c_1 \end{bmatrix}$$
4. Since they satisfy equation 7.1.1, $\lambda$ and $v$ are an eigenvalue/eigenvector pair for this matrix $A$.
