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Let $A,B$ be $n \times n$ matrices and $C,D$ be $n \times 1$ matrices. Moreover, all entries of $A,B,C,D$ are non-negative. Assume that there is a unique matrix $X$ that solves $AX = C$.

My goal is not to calculate $X$, but to determine which entry of $BX - D$ is non-negative. In solving my problem, I need to repeat this procedure several times.

I would like to ask if there is an efficient method (or references) to do so. Thank you so much!

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  • $\begingroup$ Make your life easier and call vectors "vectors" rather than "$n\times 1$ matrices" :-) $\endgroup$ May 29, 2020 at 21:57
  • $\begingroup$ Hi @WolfgangBangerth, I've just posted the refinement of this question here. Because this is an essential part of my thesis, I hope that you have a look at it and lend me some suggestion ^_^ $\endgroup$
    – Akira
    May 29, 2020 at 22:00
  • $\begingroup$ I have no idea about what can or can not be shown in this area. $\endgroup$ May 30, 2020 at 15:07
  • $\begingroup$ Thank you for your response @WolfgangBangerth ^^ $\endgroup$
    – Akira
    May 30, 2020 at 15:08

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