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Consider a linear system, $Ax=b$ where $A$ is the coefficient matrix, $x$ is the unknown vector of variables, and $b$ is a vector of constants in which all entries are same.

Is there a way to find out relative values of $x$ without actually solving the system of equations? In other words, can we say by just looking at $A$ that $x_1$ will be greater than $x_2$ because the sum of entries of first row is greater than the sum of entries of the 2nd row of $A$.

I have intentionally made the assumption that all entries of $b$ are same, so that they have no effect on our approximate solution.

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  • $\begingroup$ If you have $A x = b$ and you would like to check if the solution satisfies, say, $x_1 \geq x_2 \geq \cdots \geq x_n$, then you can use linear programming. $\endgroup$ – Rodrigo de Azevedo Mar 12 '18 at 12:29
  • $\begingroup$ Why don't you want to solve the system? $\endgroup$ – rviertel Mar 12 '18 at 19:40
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A counterexample of what you propose: $$A = \left[\begin{matrix} 1 & 1&0\\0 & 1& 0\\ 0 & 0 &1\end{matrix}\right], \quad b=\left[\begin{matrix}1\\1\\1\end{matrix}\right]$$

The sum of he first row of $A$ is greater than the rest and $x_1=0<x_2=x_3=1$

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  • $\begingroup$ your example proves that the sum of entries in a row cannot be used to find approximate values of x. Is there any other method which I can use? Again I only know A and b. $\endgroup$ – user19180 Mar 12 '18 at 10:46
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    $\begingroup$ Solving the system... $\endgroup$ – HBR Mar 12 '18 at 10:51

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