$\left | A - \lambda I \right | =0$
$\Rightarrow \begin{vmatrix} 1 - \lambda & 2 & 3 & 4 & 5 \\ 5 & 1 - \lambda & 2 & 3 & 4 \\ 4 & 5 & 1 - \lambda & 2 & 3\\ 3 & 4 &5 & 1 - \lambda &2 \\ 2 & 3 & 4 & 5 & 1 - \lambda \end{vmatrix} = 0$
Perform the row operation: $ R_1 \rightarrow R_1 + R_2 + R_3 + R_4 + R_5 $
$\Rightarrow \begin{vmatrix}
15 - \lambda & 15 - \lambda & 15 - \lambda & 15 - \lambda & 15 - \lambda \\
5 & 1 - \lambda & 2 & 3 & 4 \\
4 & 5 & 1 - \lambda & 2 & 3\\
3 & 4 &5 & 1 - \lambda &2 \\
2 & 3 & 4 & 5 & 1 - \lambda
\end{vmatrix} = 0 $
So, $ \lambda = 15 $ is the eigen value.