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I have a matrix equation $Ax=b$, where $A$ is a $4\times4$ non-singular M-matrix ($A$ has negative off-diagonal and positive diagonal entries) and $b$ is a strictly positive vector.

Let $x=(x_1, x_2, x_3, x_4)^T$ be the solution to this system of equations. Is it possible to characterize properties of $A,b$ for which $x_1<x_2<x_3<x_4$?

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If the positive diagonal entries are much larger compared to the off-diagonal negative entries, then your matrix is "almost" diagonal and so you can write down approximate conditions basically treating $A$ as diagonal and then writing down the inequalities fof $x_i$ entries in terms of the entries of $A^{-1} b$. If you treat $A$ as diagonal, it is straightforward to get the joint conditions on entries of $A$ and $b$ that make your desired inequalities hold.

Without such a "dominating diagonal" condition, the relative magnitudes of both the positive diagonal entries as well as the negative off-diagonal entries will dictate everything ($A$ could even be arbitrarily close to singular), and since the solution for $x$ is $A^{-1}b$, it will be very hard to write down any simple concrete conditions to make your desired inequalities hold, other than, say, using Cramer's rule to get the solution for $x$ explicitly as rational functions in terms of the entries of $A$ and $b$, and then writing down the inequalities in terms of those rational functions.

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