# Math Help - Linear Algebra understanding question

1. ## Linear Algebra understanding question

find the vector $\vec{v}\in \mathbb{R}^3$
who meets the requirements:

for every matrix $A=\begin{bmatrix}
a_1_1 & a_1_2 & a_1_3\\
a_2_1 & a_2_2 & a_2_3\\
a_3_1 & a_3_2 & a_3_3
\end{bmatrix}$

exist:
$A\vec{v}=\begin{bmatrix}
a_1_2-a_1_3 \\
a_2_2-a_2_3 \\
a_3_2-a_3_3
\end{bmatrix}$

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

2. Define your vector to be
$
\vec{v}=\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
$

And look at the value of $A\vec{v}$ algebraically. You should be able to get exactly what $\vec{v}$ should be.

For example, the first row of $A\vec{v}$ should be $a_{11}x + a_{12}y + a_{13}z$, and this is equal to $a_{12} - a_{13}$.
What are $x, y, z$?

3. it seems to me that is should be $\begin{bmatrix}
0\\
1\\
-1
\end{bmatrix}$

am i right?

and i cold just write it like that?

4. Yes, that is the only possible vector for the solution.

If you are unsure pick values for A that make A invertible (easiest would be the identity matrix).

Now substituting values, we have:

$
\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}
\vec{v}=\begin{bmatrix}
0 - 0 \\
1 - 0 \\
0 - 1
\end{bmatrix}
$

Which gives you the unique solution.

5. Ok i got it!
Thank you very much!