Some $3\times 3$ matrix $A$ has eigenvalue $\lambda = -2$ repeated twice.
The matrix $A$ has a $-2$-eigenvector $\boldsymbol{\vec{v}} = \left\lbrack\begin{array}{c} 1 \\ -2 \\ 2 \end{array}\right\rbrack$ and $\boldsymbol{\vec{v}}$ has a generalized eigenvector $\boldsymbol{\vec{w}} = \left\lbrack\begin{array}{c} 3 \\ -2 \\ 2 \end{array}\right\rbrack$.

A. Modify $\boldsymbol{\vec{v}}$ to get two other vectors which are also $-2$-eigenvectors:

$\qquad \boldsymbol{\vec{v}}' =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

and     $\boldsymbol{\vec{v}}'' =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

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B. Modify $\boldsymbol{\vec{w}}$ to get two other vectors which also generalize $\boldsymbol{\vec{v}}$:

$\qquad \boldsymbol{\vec{w}}' =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

and     $\boldsymbol{\vec{w}}'' =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

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C. Now write a vector which generalizes the $-2$-eigenvector $\left\lbrack\begin{array}{c} 2 \\ -4 \\ 4 \end{array}\right\rbrack$

$\qquad \phantom{\boldsymbol{\vec{w}'} = }$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

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If you don't get this in 10 tries, you can get a hint.

Hint: