Properties of a square matrix whose rows and columns sum to form vectors orthogonal to the "all ones" vector?

Question

I am interested in a type of matrix, $A$, with the following characteristics:

• $A$ is a square $n \times n$ matrix
• Summing $A$ along rows, produces an $n \times 1$ vector that is orthogonal to the "all ones" vector i.e. an $n \times 1$ whose elements are each equal to $1$.
• Summing $A$ along columns, produces an $1 \times n$ vector that is orthogonal to the "all ones" vector i.e. an $1 \times n$ whose elements are each equal to $1$.
• $A$ is real
• The sum of all elements of $A$ is equal to zero (This a consequence of bullet points 2 & 3)
• $Rank(A) = 3$
• $A$ is symmetric
• $|A_{ij}| \leq 1 $

Are there any special properties of matrices like this?

Answer 1

Every matrix $A$ whose elements sum to zero will be a solution.
The dot-product with the all-ones vector gives the sum of the entries in a vector. Each entry in the vector was the sum of a column (or a row), so the dot-product gives the sum of all the entries in the matrix.