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A 19x19 matrix of 1 and -1

Source: INAMO 2003 - Problem 4

August 5, 2011

linear algebramatrixcombinatorics unsolvedcombinatorics

Problem Statement

Given a

19 \times 19

matrix where each component is either

1

or

-1

. Let

b_i

be the product of all components in the

i

-th row, and

k_i

be the product of all components in the

i

-th column, for all

1 \le i \le 19

. Prove that for any such matrix,

b_1 + k_1 + b_2 + k_2 + \cdots + b_{19} + k_{19} \neq 0

.

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