find the vector $\displaystyle \vec{v}\in \mathbb{R}^3$

who meets the requirements:

for every matrix $\displaystyle A=\begin{bmatrix}

a_1_1 & a_1_2 & a_1_3\\

a_2_1 & a_2_2 & a_2_3\\

a_3_1 & a_3_2 & a_3_3

\end{bmatrix}$

exist:

$\displaystyle A\vec{v}=\begin{bmatrix}

a_1_2-a_1_3 \\

a_2_2-a_2_3 \\

a_3_2-a_3_3

\end{bmatrix}$

Can some one please help me here? is $\displaystyle \vec{v}$ is eigenvector?!

Thank you!