If the sum of the elements of an inverse of a matrix $A$ is $0$, does it affect somehow the determinant or something else of the matrix $A$, where $A$ is a $3\times 3$ matrix?
I tried for a $2\times 2$ matrix, and I get that also the sum of the elements of the matrix A is also $0$, but I do not know if it applies for any $n\times n$ matrix.
This is not even true for a $2 \times 2$ matrix, take $$A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}, \quad A^{-1} = \begin{pmatrix} -3 & 2 \\ 2 & -1 \end{pmatrix}$$