Problem 5, 2013

Let \(n \geq 2\) be a natural number. Every cell of a square table \(A\) of size \(n \times n\) is filled with one of the numbers \(1\) and \(-1\). For each \(i \in \{1, 2, \ldots, n\}\) write \(k_i\) for the product of the numbers in the \(i\)-th column and \(v_i\) for the product of the numbers in the \(i\)-th row. The table is called perfect if

Determine all natural numbers \(n \geq 2\) for which a perfect table exists.