8. Let $n\ge3$. Let $A=((a_{ij}))_{1\le i,j\le n}$ be an $n×n$ matrix such that $a_{ij}\in\{1,-1\}$ for all $1\le i,j\le n$. Suppose that $$a_{k1}=1\quad \forall1\le k\le n$$ $$\sum_{k=1}^na_{ki}a_{kj}=0\quad\forall i\ne j.$$ Show that $n$ is a multiple of $4$.
Yesterday this question was asked in an examination. I could not solve this problem but intuitively this is correct. A proof is needed.
