OSN 2003 Problem 4

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