A matrix maths problem

Apr 2018
3
0
UK
I've been given this as part of my degree and I'm struggling to solve it:

Suppose A is a 10x10 matrix with integer entries, with the following property: given any five rows and five columns, the sum of the entries of the 5x5 matrix formed by these rows and columns is even. Prove that all the entries of A are even.

I've tried proving it first for any 1 row and 5 columns (forming a 1x5 matrix) so that I can then possibly use induction but I'm still struggling.

Any help would be greatly appreciated! Thanks!
 
Jan 2017
6
3
Banja Luka
I don't have any solutions, but perhaps you could try to assume the opposite. That there exists one or more odd entries and that the sum of the matrix produced by the method you described is even, and then try to land on a contradiction. Just an idea.
 
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SlipEternal

MHF Helper
Nov 2010
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1,571
You have 100 variables (the entries of the 10x10 matrix). You have $\dbinom{10}{5}^2$ equations of the form:
$$\sum_{i=1}^5\sum_{j=1}^5 a_{(r_i,c_j)}\cong 0\pmod{2} $$

If there are more than 100 of these equations that are linearly independent then the trivial solution is the only solution. Which over integers mod 2 means every entry must be even.
 
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SlipEternal

MHF Helper
Nov 2010
3,728
1,571
You have 100 variables (the entries of the 10x10 matrix). You have $\dbinom{10}{5}^2$ equations of the form:
$$\sum_{i=1}^5\sum_{j=1}^5 a_{(r_i,c_j)}\cong 0\pmod{2} $$

If there are more than 100 of these equations that are linearly independent then the trivial solution is the only solution. Which over integers mod 2 means every entry must be even.
Oops, I mean at least 100 of these equations that are linearly independent. There cannot be more than 100 of them linearly independent.