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 that this $3\times3$ matrix should have such that row sum(sum of entries of a particular row will give the row sum of that row) of inverse of this matrix is 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 may be there is some kind of other property(preferably having some physical significance like values are increasing or something like this) which will result in positive entries of the matrix.
Best Answer
$\boxed{\text{HINT}}$
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.