Chapter 7 - Eigenvalues and Eigenvectors - 7.1 Eigenvalues and Eigenvectors - 7.1 Exercises - Page 350: 3

See below.

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