1965 IMO 第 2 题

Consider the system of equations $$\left\{\begin{array}{l} a_{11} x_{1}+a_{12} x_{2}+a_{13} x_{3}=0 \\ a_{21} x_{1}+a_{22} x_{2}+a_{23} x_{3}=0 \\ a_{31} x_{1}+a_{32} x_{2}+a_{33} x_{3}=0 \end{array}\right.$$ whose coefficients satisfy the following conditions: (a) $a_{11}, a_{22}, a_{33}$ are positive real numbers; (b) all other coefficients are negative; (c) in each of the equations the sum of the coefficients is positive. Prove that $x_{1}=x_{2}=x_{3}=0$ is the only solution to the system.

考虑方程组 \left\{\begin{array}{l} 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 \end{array}\right。的系数满足以下条件： (a) $a_{11}、a_{22}、a_{33}$为正实数； (b) 所有其他系数均为负； (c) 在每个方程中，系数之和为正。证明 $x_{1}=x_{2}=x_{3}=0$ 是系统的唯一解。