4

Part of 2003 Indonesia MO

Problems(1)

A 19x19 matrix of 1 and -1

Source: INAMO 2003 - Problem 4

19 \times 19

matrix where each component is either

1

-1

b_i

be the product of all components in the

i

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

linear algebramatrixcombinatorics unsolvedcombinatorics