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        riddles >> putnam exam (pure math) >> Magic Matrix Square
        
(Message started by: JocK on May 22nd, 2006, 1:13pm)
Title: Magic Matrix Square
Post by JocK on May 22nd, 2006, 1:13pm
Find nine nxn matrices Ai,j such that when placed in a 3x3 square:

A1,1  A1,2  A1,3
A2,1  A2,2  A2,3
A3,1  A3,2  A3,3  

the product of the matrices in a row equal the unit matrix:

Ak,1 Ak,2 Ak,3  =  1  

(k = 1,2,3), whilst the product of the matrices in a column yield minus the unit matrix:

A1,k A2,k A3,k  =  -1

What is the smallest matrix dimension n for which such a magic square can be formed?


 
Title: Re: Magic Matrix Square
Post by Sjoerd Job Postmus on May 22nd, 2006, 1:27pm

on 05/22/06 at 13:13:18, JocK wrote:
Find nine nxn matrices Ai,j such that when placed in a 3x3 square:

A1,1  A1,2  A1,3
A2,1  A2,2  A2,3
A3,1  A3,2  A3,3  

the product of the matrices in a row equal the unit matrix:

Ak,1 Ak,2 Ak,3  =  1  

(k = 1,2,3), whilst the product of the matrices in a column yield minus the unit matrix:

A1,k A2,k A3,k  =  -1

What is the smallest matrix dimension n for which such a magic square can be formed?


 

Wouldn't
[hideb]-1 -1 -1
-1 -1 -1
1   1   1[/hideb]
satisfy the equation?
At least, I assume [hide]-1 * -1 = 1[/hide]

So [hide]-1 * -1 * 1 = 1[/hide]
And [hide]-1 * -1 * -1 = -1[/hide]

Or maybe I'm being too obvious ;)
Title: Re: Magic Matrix Square
Post by JocK on May 22nd, 2006, 2:13pm

on 05/22/06 at 13:27:21, Sjoerd Job Postmus wrote:
Wouldn't
-1 -1 -1
-1 -1 -1
1   1   1
satisfy the equation?


All columns lead to a product +1, and most rows lead to a product -1. However, not all rows do (the last row doesn't) ...
Title: Re: Magic Matrix Square
Post by Grimbal on May 24th, 2006, 6:04am
[hideb]+------+------+------+
| 1  0 |-1  0 |-1  0 |
| 0  1 | 0  1 | 0  1 |
+------+------+------+
| 0 -1 | 1  0 | 0  1 |
| 1  0 | 0  1 |-1  0 |
+------+------+------+
| 0 -1 | 1  0 | 0  1 |
| 1  0 | 0 -1 | 1  0 |
+------+------+------+
[/hideb]
Title: Re: Magic Matrix Square
Post by JocK on May 24th, 2006, 11:56am
Well done, ... but ....

... within the third row (and within the third column) not all matrices commute. Hence, the corresponding row-product (column-product) is not unambiguously defined.

Can you generate a realisation such that within each row and each column all matrices commute...?

(Sorry, should have stated this requirement more clearly.)


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