This example is given in Higham [2002], and is provided without explanation. I am not sure how the condition number of the matrix is just 5. How can you directly calculate the condition number of a matrix with epsilon? I know that cond(A) = $\left\Vert |A^{-1}||A| \right\Vert_{\inf}$, but I'm not sure how that helps here.
I calculated $A^{-1} = \{(1, -1/\epsilon, 1),(0, 1/\epsilon, -1), (0, 0, 1)\}$
$|T| = \begin{bmatrix} 1 & 1 & 0 \\ 0 & |\epsilon| & |\epsilon| \\ 0 & 0 & 1 \end{bmatrix} $
$|T^{-1}| = \begin{bmatrix} 1 & {1 \over |\epsilon|} & 1 \\ 0 & {1 \over |\epsilon|} & 1 \\ 0 & 0 & 1\end{bmatrix}$
$|T||T^{-1}| = \begin{bmatrix} 1 & 2 & 2 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}$
$\| |T||T^{-1}| \|_\infty = 5$.