8th Putnam 1948

Problem B6

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 u2 + v2 + w2 = 0.

(2) Let (aij) be an n x n matrix. Suppose that for each i, 2 |aii| > ∑1n |aij|. By considering the corresponding system of linear equations or otherwise, show that det aij ≠ 0.




(2) If the det is non-zero, then we can find xinot all zero so that ∑ xi aij = 0 for each j. Take k so that |xk| ≥ |xi| for all i. Then |∑i not k xiaik| ≤ ∑i not k |xi aik| ≤ |xk| ∑i not k |aik| < |xk| |akk|, so we cannot have ∑ xiaik = 0. Contradiction.



8th Putnam 1948

© John Scholes
11 Mar 2002