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!