I am interested in $n \times n$ matrices over some field $K$ all whose rows and all whose columns sum to zero.
First question: do these matrices have a name?
Pending an answer I will call these "null-matrices".
Second (main) question:
Given $n$, are there subsets $J \subset \{1, \ldots n\} \times \{1, \ldots, n\}$ of indices such that
$$\sum_{(i, j) \in J} a_{i,j} = 0$$
for every null-matrix $(a_{ij})_{i, j = 1 \ldots n}$?
More visually: can you take a red pencil and put red circles around some entries in a an (still empty) matrix so that in whichever way someone fills up the matrix with elements of $K$ to obtain a null-matrix, the sum of the red-circled entries will always add up to zero?
Obviously the answer is yes, just take $J$ to be a disjoint union of rows or a disjoint union of columns. So my question is: are there examples of set $J$ that are not of this form?
In general (i.e. without specifying the field over which we consider the matrix) I expect the answer to be no (but please prove me wrong) – however for some special combinations of $n$ and char($K$) the answer might be yes. In particular, for $n = 2$, char($K$) $=2$, the diagonal (i.e. $J = \{(1, 1), (2,2)\}$) is an example of the type of set I'm looking for.
Are there more examples like this?
Best Answer
If $2=0$, examples can be constructed by partitioning the index set for the rows and columns into two subsets $A$ and $B$, and taking the subset $(A \times A) \cup (B \times B)$.
No nontrivial examples exist in characteristic $\neq 2$. Considering matrices that are $0$ outside a $2 \times 2$ minor shows that if a subset meets the conditions and contains two positions of the matrix that do not share a row or columns, it has to contain all $4$ corners of the $2 \times 2$ minor with those two points as a diagonal. Therefore, the subset is a Cartesian product of a subset of rows with a subset of columns. The only such sets with guaranteed zero sum of entries are unions of several complete rows, or of several complete columns.