AT-OEMO-FED1 2007 Problem 1

In a quadratic table with $2007$ rows and $2007$ columns is an odd number written in each field. For $1\leq i\leq2007$ is $Z_i$ the sum of the numbers in the $i$ -th row and for $1\leq j\leq2007$ is $S_j$ the sum of the numbers in the $j$ -th column. $A$ is the product of all $Z_i$ and $B$ the product of all $S_j$ . Show that $A+B\neq0$