Let \( A = [a_{ij}] \) for \( i, j = 1, 2, 3, \ldots, n \) be an \( n \times n \) matrix such that
\( a_{ij} = \begin{cases} 1, & \text{if } (i+j) \text{ is even} \\ -1, & \text{if } (i+j) \text{ is odd} \end{cases} \)
where \( n \gt 1 \). Then the rank of \( A \) is: