IMO 1965 Problem 2

Consider the sytem of equations $$a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0$$$$a_{21}x_1+a_{22}x_2+a_{23}x_3 =0$$$$a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0$$ with unknowns $x_1, x_2, x_3$ . The coefficients satisfy the conditions:

a) $a_{11}, a_{22}, a_{33}$ are positive numbers;

b) the remaining coefficients are negative numbers;

c) in each equation, the sum ofthe coefficients is positive.

Prove that the given system has only the solution $x_1=x_2=x_3=0$ .