Answer
See below
Work Step by Step
There exists a matrix $S =\begin{bmatrix}
v_1,v_2,v_3
\end{bmatrix}$ such that $S^{-1}AS=\begin{bmatrix}
\lambda & 1 & 0\\
0 & \lambda & 1 \\
0 & 0 & \lambda
\end{bmatrix} \\
\rightarrow A[v_1,v_2,v_3]=[v_1,v_2,v_3] \begin{bmatrix}
\lambda & 1 & 0\\
0 & \lambda & 1 \\
0 & 0 & \lambda
\end{bmatrix} \\
\rightarrow Av_1=\lambda v_1\\
Av_2=v_1+\lambda v_2\\
Av_3=v_2+\lambda v_3 \\
\rightarrow (A- \lambda1)v_1=0\\
(A-\lambda 1) v_2=v_1 \\
(A- \lambda 1)v_3=v_2$
