MathDB

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.

B6