Jordan block/Power/Effect on 11.../Exercise

Show the equality

${\displaystyle {}{\begin{pmatrix}\lambda &1&0&\cdots &0\\0&\lambda &1&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\vdots &\cdots &0&\lambda &1\\0&\cdots &\cdots &0&\lambda \end{pmatrix}}^{n}{\begin{pmatrix}1\\1\\0\\\vdots \\0\end{pmatrix}}={\begin{pmatrix}\lambda ^{n}+n\lambda ^{n-1}\\\lambda ^{n}\\0\\\vdots \\0\end{pmatrix}}\,.}$