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 $x>y>z$? Everything is in $\mathbb{R}$.

Any reference would be great!

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!