Existence of Matrices with Prescribed Off-Diagonal Block Element Sums

Necessary and sufficient conditions are proven for the existence of a square matrix, over an arbitrary field, such that for every principal submatrix the sum of the elements in the row complement of the submatrix is prescribed. The problem is solved in the cases where the positions of the nonzero elements of A are contained in a given set of positions, and where there is no restriction on the positions of the nonzero elements of A. The uniqueness of the solution is studied as well. The results are used to solve the cases where the matrix is required to be symmetric and/or nonnegative entrywise.