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Topic: Magic Matrix Square (Read 497 times) |
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JocK
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Magic Matrix Square
« on: May 22nd, 2006, 1:13pm » |
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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?
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| « Last Edit: May 22nd, 2006, 1:16pm by JocK » |
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Sjoerd Job Postmus
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Posts: 228
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Re: Magic Matrix Square
« Reply #1 on: May 22nd, 2006, 1:27pm » |
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on May 22nd, 2006, 1:13pm, 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?
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Wouldn't
| hidden: | -1 -1 -1
-1 -1 -1
1 1 1 |
satisfy the equation?
At least, I assume -1 * -1 = 1
So -1 * -1 * 1 = 1
And -1 * -1 * -1 = -1
Or maybe I'm being too obvious 
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JocK
Uberpuzzler
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Posts: 877
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Re: Magic Matrix Square
« Reply #2 on: May 22nd, 2006, 2:13pm » |
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on May 22nd, 2006, 1:27pm, Sjoerd Job Postmus wrote:
Wouldn't
-1 -1 -1
-1 -1 -1
1 1 1
satisfy the equation?
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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) ...
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Grimbal
wu::riddles Moderator
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Re: Magic Matrix Square
« Reply #3 on: May 24th, 2006, 6:04am » |
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| hidden: | +------+------+------+
| 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 |
+------+------+------+
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JocK
Uberpuzzler
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Posts: 877
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Re: Magic Matrix Square
« Reply #4 on: May 24th, 2006, 11:56am » |
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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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IP Logged |
solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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