*Do either (1) or (2):*

(1) Take the origin O of the complex plane to be the vertex of a cube, so that OA, OB, OC are edges of the cube. Let the feet of the perpendiculars from A, B, C to the complex plane be the complex numbers u, v, w. Show that u^{2} + v^{2} + w^{2} = 0.

(2) Let (a_{ij}) be an n x n matrix. Suppose that for each i, 2 |a_{ii}| > ∑_{1}^{n} |a_{ij}|. By considering the corresponding system of linear equations or otherwise, show that det a_{ij} ≠ 0.

**Solution**

(1)

(2) If the det is non-zero, then we can find x_{i}not all zero so that ∑ x_{i} a_{ij} = 0 for each j. Take k so that |x_{k}| ≥ |x_{i}| for all i. Then |∑_{i not k} x_{i}a_{ik}| ≤ ∑_{i not k} |x_{i} a_{ik}| ≤ |x_{k}| ∑_{i not k} |a_{ik}| < |x_{k}| |a_{kk}|, so we cannot have ∑ x_{i}a_{ik} = 0. Contradiction.

© John Scholes

jscholes@kalva.demon.co.uk

11 Mar 2002