Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix} $$ All the elements of A are positive. Now I want to know if there are any properties of this $3\times3$ matrix necessary for the row sums (given by the sum of a particular row's entries) of the inverse of this matrix are always positive.
I tried to take increasing values in the columns, i.e. $a_{11} > a_{21} > a_{31}$, and similarly for all the columns but it depends majorly on the magnitude. So for different magnitude it came out to be different. I also tried for decreasing values but did not reach anywhere. So I thought maybe there is some kind of other property (preferably with some physical significance like values are increasing or something like this) which will result in positive entries of the matrix.
$\boxed{\text{HINT}}$
Let $$\sum_k A_{ik}B_{kj}=\delta_{ij}$$
($B$ is the inverse of $A$). We have: $$\sum_k A_{ik}\sum_jB_{kj}=\sum_k\sum_jA_{ik}B_{kj}=\sum_j\delta_{ij}=1$$
Now you have a relation between the row sum of the matrix and that of the inverse.