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

See below.

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