MathDB

Let

n\geqslant 3

. Let

A=((a_{ij}))_{1\leqslant i,j\leqslant n}

be an

n\times n

matrix such that

a_{ij}\in\{-1,1\}

for all

1\leqslant i,j\leqslant n

. Suppose that

a_{k1}=1~~\text{for all}~1\leqslant k\leqslant n

and

~~\sum_{k=1}^n a_{ki}a_{kj}=0~~\text{for all}~i\neq j

.
Show that

n

is a multiple of

4

8