Can this be done in a simplier way? Magic Squares

[Dustinsfl]

Proving Axiom 1 of all 3 x 3 magic squares.

I used summation notation to do so but it is extremely long and cumbersome.

I attached the pdf file with the work.

Is there a way to do this in a simpler more concise manner?

 

[Dustinsfl]

I just added the pdf file I forgot to add.

 

[Tinyboss]

What is this axiom 1 that you're trying to prove? That PDF is impenetrable.

 

[Dustinsfl]

Impenetrable? https://www.physicsforums.com/showthread.php?t=384823&page=2

If you go to this link, you can catch up what we (the forum) have done. I need to form the basis for S but I am not sure where to start.

 

[Zorba]

To form the basis, consider the equations you have regarding some $$A$$ a magic square, denote $$x_i = a_{1i} \in A \quad 1 \le i \le 3$$ and $$x_i = a_{2i} \in A \quad 4 \le i \le 6$$ and so on, then:

$$x_1 + x_2 + x_3 = x_4 + x_5 + x_6$$
$$x_1 + x_2 + x_3 = x_7 + x_8 + x_9$$
$$x_1 + x_4 + x_7 = x_2 + x_5 + x_8$$
$$x_1 + x_4 + x_7 = x_3 + x_6 + x_9$$
$$x_1 + x_2 + x_3 = x_1 + x_4 + x_7$$

If you play with these equations for a bit you can get the basis matrices.