Vampire Matrices algoritm

Can you find two 3x3 matrices L and R with integer values such that $LR = nL+R$ with $n$ an integer constant?
First, let write our matrices as

$L = \begin{bmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\\ \end{bmatrix}\\R = \begin{bmatrix} x_1 & y_1 & z_1\\ x_2 & y_2 & z_2\\ x_3 & y_3 & z_3\\ \end{bmatrix}$
This means that $LR = \left[\begin{matrix}a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} & a_{1} y_{1} + a_{2} y_{2} + a_{3} y_{3} & a_{1} z_{1} + a_{2} z_{2} + a_{3} z_{3}\\b_{1} x_{1} + b_{2} x_{2} + b_{3} x_{3} & b_{1} y_{1} + b_{2} y_{2} + b_{3} y_{3} & b_{1} z_{1} + b_{2} z_{2} + b_{3} z_{3}\\c_{1} x_{1} + c_{2} x_{2} + c_{3} x_{3} & c_{1} y_{1} + c_{2} y_{2} + c_{3} y_{3} & c_{1} z_{1} + c_{2} z_{2} + c_{3} z_{3}\end{matrix}\right]$
And $nL+R = \left[\begin{matrix}a_{1} n + x_{1} & a_{2} n + y_{1} & a_{3} n + z_{1}\\b_{1} n + x_{2} & b_{2} n + y_{2} & b_{3} n + z_{2}\\c_{1} n + x_{3} & c_{2} n + y_{3} & c_{3} n + z_{3}\end{matrix}\right]$
Combining these to expressions with $LR = nL+R$ yields nine equations of a similar form.
$\alpha_1 \gamma_1 + \alpha_2 \gamma_2 + \alpha_3 \gamma_3 = n\alpha_i + \gamma_j$

Now, let us define a mathamatical object, that we call a local triple, that has the following properties.

1. A 3d vector $(\alpha_1,\alpha_2,\alpha_3)$ of integers
2. An index 1,2,3 related to the row/column that the triple is on
3. A token, left/right, based on if the triple is in the left or right matrix.
   We will note left local triples as $\left(\alpha_1,\alpha_2,\alpha_3, j\right|$ and right triples as $\left|\gamma_1,\gamma_2,\gamma_3, i\right)$
   Let us call two triples $\left(\alpha_1,\alpha_2,\alpha_3, j\right|$ and $\left|\gamma_1,\gamma_2,\gamma_3, i\right)$connected if $\alpha_1 \gamma_1 + \alpha_2 \gamma_2 + \alpha_3 \gamma_3 = n\alpha_i + \gamma_j$
   Note that the index of a local triple indexes the vector of the other local triple, not the own vector.
   Now that this is done, we can check for each left local triple all possible right local triples, and store them in a set.
   Then we can check for each set of 3 left local triples, with each a different index, if the intersection for the three corresponding sets contains at least one right local triple of each index. If it does, we can use the 6 local triples with corresponding indexes to build our original matrices back.
   The rust code that performs this for single digit non zero integers for $n=10$ can be found here: https://gitlab.com/Elmosfire/vampire-matrices/-/blob/master/main.rs