Chapter 7 - Eigenvalues and Eigenvectors - 7.1 Eigenvalues and Eigenvectors - 7.1 Exercises - Page 350: 9

See below.

Because $\begin{bmatrix} 1& 1 \\ 1& 1\\ \end{bmatrix} \begin{bmatrix} c \\ -c\\ \end{bmatrix} =\begin{bmatrix} 0 \\ 0\\ \end{bmatrix} =0 \begin{bmatrix} c \\ -c\\ \end{bmatrix} $ and $\begin{bmatrix} 1& 1 \\ 1& 1\\ \end{bmatrix} \begin{bmatrix} c \\ c\\ \end{bmatrix} =\begin{bmatrix} 2c \\ 2c\\ \end{bmatrix} =2 \begin{bmatrix} c \\ c\\ \end{bmatrix} $ we have proved what we wanted to.