Problem 2

Consider the system

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

. The diagonal coefficients

a_{11},a_{22},a_{33}

are positive, the other coefficients are negative, and the sum of the coefficients in each equation is positive. Prove that the system has only the solution

x_1=x_2=x_3=0

Step 3 of 4: Contradict cancellation

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|a_{11}x_1|> |a_{12}x_2|+|a_{13}x_3|.

Detailed analysis

By strict diagonal dominance and maximality, $a_{11}|x_1|>(|a_{12}|+|a_{13}|)|x_1|\\ge |a_{12}x_2|+|a_{13}x_3|$ . Hence the first equation cannot sum to zero, contradicting that it is a solution.