Magic Square Matrix

Date: 03/24/2003 at 10:47:27
From: John
Subject: Magic squares

Let M be an integer-valued 3x3 matrix whose entries form a magic 
square. Let s be the sum of all entries in M and d be the determinant 
of M. Show that d/s is an integer.

Date: 03/28/2003 at 07:46:54
From: Doctor Jacques
Subject: Re: Magic squares

Hi John,

We want to show that d = 0 (mod 3x).

Without changing d, we can add the left and right columns to the 
middle one, and do the same with the rows. This yields a matrix:

  | a  x  c |
  | x  0  x |
  | g  x  i |

(I replaced 3x by 0, since we are interested in the value mod 3x).

If you remember that in a magic square, a + e + i = g + e + c, you 
should be able to complete the proof.

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/31/2003 at 05:53:41
From: john
Subject: Thank you (magic squares)

Doctor Jacques,

Thank you, that was a great help. Although I adapted your 
advice a little (only by leaving the centre value as 3x, 
and proceeding from there), the method was perfect, and 
relatively simple compared to what I had been trying. 
Thanks again.