2 problems - Matrix numbering and integer colouring

Here are 2 interesting problems from chapter 2 of Arthur Engel's book:

1. The positive integers are colored black and white. The sum of two differently colored
numbers is black, and their product is white. What is the product of two white
numbers? Find all such colorings.

2. Each element of a 25 × 25 matrix is either +1 or −1. Let ai be the product of all
elements of the i-th row and bj be the product of all elements of the j-th column.
Prove that a₁ + b₁ + · · · + a₂₅ + b₂₅ ≠ 0.