[math-fun] sum-product matrices

20 Jan 2010

      A matrix of distinct positive integers is sum-product if its rows have equal 
sum and columns have equal (possibly distinct) product.

Any 1 x 1 matrix is trivially sum-product.

No 1 x n or n x 1 matrix is sum-product.

No 2 x 2 matrix is sum-product. In a sum-product matrix

a b

c d

we must have a+b = c+d and ac = bd. But then ac+bc = bc+bd => (a+b)c = 
b(c+d) => c = b, and the elements are not distinct.

There are an infinitude of 2 x 3 sum-product matrices. I believe the 
smallest, both in row sum and colum product, is

4 10 12

15 6 5

with row sum 26 and column product 60.

I believe the following is true:

For positive integers x, y, z, let u = x+y+z and v = xy+xz+yz.

Then the matrix M(x, y, z) =

vx vy vz

uyz uxz uxy

has equal row sums uv and equal column products xyzuv.

If you choose x < y < z with xz < y^2 and gcd(x, y, z) = 1, form M(x, y, z), 
and divide this by the gcd of its elements, then I believe you will generate 
the "primitive" 2 x 3 sum-product matrices.

I'm pretty sure that apart from 1 x 1 matrices, there are no square 
sum-product matrices.

?

But I was pleased to find that there are indeed sum-product matrices with 
equal row sum and column product. The following are (all?) "primitive" 14 x 
2 sum-product matrices with sum = product = 1890. By "primitive" I mean 
distinct up to row and column permutation.

2 3 15 45 63 70 90 105 135 189 210 270 315 378
945 630 126 42 30 27 21 18 14 10 9 7 6 5

2 5 7 21 30 45 70 105 126 135 189 210 315 630
945 378 270 90 63 42 27 18 15 14 10 9 6 3

2 5 9 14 27 45 63 90 105 126 189 270 315 630
945 378 210 135 70 42 30 21 18 15 10 7 6 3

2 5 14 15 18 27 42 63 90 189 210 270 315 630
945 378 135 126 105 70 45 30 21 10 9 7 6 3

2 6 7 18 27 30 42 90 126 135 189 210 378 630
945 315 270 105 70 63 45 21 15 14 10 9 5 3

2 7 9 15 18 27 30 45 90 135 189 315 378 630
945 270 210 126 105 70 63 42 21 14 10 6 5 3

David Wilson

Erich Friedman

David Wilson

Christian Boyer

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