Answer
See below
Work Step by Step
1. According to equation 7.1.1:
$$Av = \lambda v$$
2. Calculate $Av$:
$$Av =
\begin{bmatrix} 6 & -1 & 0 \\ -16 & 6 & 0 \\ -4 & -1 & 10 \end{bmatrix}
\begin{bmatrix} c_1\\ -4c_1 \\ c_2 \end{bmatrix} =\begin{bmatrix} 6c_1+4c_1 \\ -16c_1- 24c_1
\\ -4c_1+4c_1+10c_2 \end{bmatrix}=\begin{bmatrix} 10c_1 \\ -40c_1
\\ 10c_2 \end{bmatrix}=10 \begin{bmatrix} c_1\\ -4c_1 \\ c_2 \end{bmatrix}
$$
3. Calculate $\lambda v$:
$$\lambda v = 10 \begin{bmatrix} c_1\\ -4c_1 \\ c_2 \end{bmatrix}$$
4. Since they satisfy equation 7.1.1, $\lambda$ and $v$ are an eigenvalue/eigenvector pair for this matrix $A$.
