In the following matrix equation, all coefficients $a_{ij}>0$ and all $a_i>0$ and the column sums in the matrix $A$ are all 0
(e.g. $-a_{11}+a_{21}+a_{31}=0$, etc.). This means that the determinant of $A$ is 0: $|A| = 0$.
$$\begin{pmatrix} -a_{11} & a_{12} & a_{13} \\ a_{21} & -a_{22} & a_{23} \\ a_{31} & a_{32} & -a_{33} \\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3\\ \end{pmatrix}= \begin{pmatrix} -a_1\\ -a_2\\ -a_3\\\end{pmatrix} $$
Conjecture:
If you now change the matrix $A$ by subtracting any value $d_i$ in the main diagonal (all $d_i>0$), the changed matrix equation only ever has a positive (i.e. all $x_i>0$) solution.
$$\begin{pmatrix} -a_{11}-d_1 & a_{12} & a_{13} \\ a_{21} & -a_{22}-d_2 & a_{23} \\ a_{31} & a_{32} & -a_{33}-d_3 \\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3\\ \end{pmatrix}= \begin{pmatrix} -a_1\\ -a_2\\ -a_3\\\end{pmatrix} $$
Question 1:
Is this conjecture correct or can it be found somewhere on the Internet as a lemma?
Question 2:
Can the conjecture also be generalized to $n\times n$ matrices ($n>3$)?
