Putnam 1948 B6

Answer wither (i) or (ii):

(i) Let $V, V_1 , V_2$ and $V_3$ denote four vertices of a cube such that $V_1 , V_2 , V_3$ are adjacent to $V.$ Project the cube orthogonally on a plane of which the points are marked with complex numbers. Let the projection of $V$ fall in the origin and the projections of $V_1 , V_2 , V_3$ in points marked with the complex numbers $z_1 , z_2 , z_3$ , respectively. Show that $z_{1}^{2} +z_{2}^{2} +z_{3}^{2}=0.$ (ii) Let $(a_{ij})$ be a matrix such that $$|a_{ii}| > |a_{i1}| + |a_{i2}|+\ldots +|a_{i i-1}|+ |a_{i i+1}| +\ldots +|a_{in}|$$ for all $i.$ Show that the determinant is not equal to $0.$