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A pretty elementary question, but does anyone know of sufficient conditions to order the solutions of a system of linear equations? For example, in the system, \begin{align*}\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{32}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix}\end{align*} are there any sufficient conditions on the $a_{ij}$ or $d_i$ such that we may say something like $x>y>z$? All components are in $\mathbb{R}$.

Any reference would be great!

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  • $\begingroup$ it will suffice for the matrix $A=(a_{ij})$ to be invertible and the coordinates of the vector $A^{-1}d$ to satisfy your inequalities $\endgroup$
    – Dima Pasechnik
    Commented Mar 28, 2023 at 22:58
  • $\begingroup$ Thanks, I agree, though would you know of any results that give conditions on $A^{-1}d$ such that we may recover these type of inequalities? Or maybe some related results? $\endgroup$
    – menritgs
    Commented Mar 29, 2023 at 8:49
  • $\begingroup$ There is nothing left to "recover" left here, the inequalities can be read off $A^{-1}d$. One further simplification is that instead of $A^{-1}$ you can use the classical adjoint (en.wikipedia.org/wiki/Adjugate_matrix) - this would allow cubic inequalities (quadratic in $a_{ij}$'s, linear in $d_k$'s - but then you'd need to consider two cases, depending on the sign of $\det A$. $\endgroup$
    – Dima Pasechnik
    Commented Mar 29, 2023 at 9:31
  • $\begingroup$ you can use en.wikipedia.org/wiki/Cramer%27s_rule - which would amount to basically the same thing. $\endgroup$
    – Dima Pasechnik
    Commented Mar 29, 2023 at 9:51
  • $\begingroup$ it's certainly a property of A and d, not A or d. $\endgroup$
    – Dima Pasechnik
    Commented Mar 30, 2023 at 0:00

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