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 2 of 4: Select a maximal coordinate

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|x_1|=\\max(|x_1|,|x_2|,|x_3|).

Detailed analysis

If a solution is nonzero, relabel the variables so that $|x_1|$ is maximal and positive. The first equation has diagonal contribution of magnitude $a_{11}|x_1|$ , while the other two contributions have total magnitude at most $(|a_{12}|+|a_{13}|)|x_1|$ .