MathLabs

Problem 2

Consider the system a11x1+a12x2+a13x3=0a_{11}x_1+a_{12}x_2+a_{13}x_3=0, a21x1+a22x2+a23x3=0a_{21}x_1+a_{22}x_2+a_{23}x_3=0, a31x1+a32x2+a33x3=0a_{31}x_1+a_{32}x_2+a_{33}x_3=0. The diagonal coefficients a11,a22,a33a_{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 x1=x2=x3=0x_1=x_2=x_3=0.
Step 4 of 4: Finish the zero case
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x1=0Longrightarrowx2=x3=0.x_1=0\\Longrightarrow x_2=x_3=0.
Detailed analysis

The contradiction rules out every nonzero solution. Therefore all three coordinates vanish, and x1=x2=x3=0x_1=x_2=x_3=0 is the unique solution.