Let $n \geq 3$. Let $A = \left( \left( a _ { i j } \right) \right) _ { 1 \leq i , j \leq n }$ be an $n \times n$ matrix such that $a _ { i j } \in \{ 1 , - 1 \}$ for all $1 \leq i , j \leq n$. Suppose that $$\begin{aligned} & a _ { k 1 } = 1 \text { for all } 1 \leq k \leq n \text { and } \\ & \sum _ { k = 1 } ^ { n } a _ { k i } a _ { k j } = 0 \text { for all } i \neq j \end{aligned}$$ Show that $n$ is a multiple of 4.