Question 2
(POL) Consider the system of equations
$$\begin{cases}a_{11}x_1+a_{12}x_2+a_{13}x_3=0,\\a_{21}x_1+a_{22}x_2+a_{23}x_3=0,\\a_{31}x_1+a_{32}x_2+a_{33}x_3=0,\end{cases}$$
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.
$$\begin{cases}a_{11}x_1+a_{12}x_2+a_{13}x_3=0,\\a_{21}x_1+a_{22}x_2+a_{23}x_3=0,\\a_{31}x_1+a_{32}x_2+a_{33}x_3=0,\end{cases}$$
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.