Suppose A is a Hadamard matrix of size $d$. Let A be in normalized in a sense that first row and first column are all ones. What is the sum of rows? I tried random Hadamard matrices and seem to get $d,0,0,0,0,\ldots$
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Follows immediately from the property of Hadamard Matrices: Any two distinct rows are orthogonal.
Since all the other rows are orthogonal to the all ones row, the sum of the elements in each of those rows must be zero.
EDIT: To answer Qiaochu's query:
MUltiplying a row or column of a Hadamard matrix by -1 gives yet another Hadamard matrix.
Using this operation, we can easily normalize a Hadamard matrix to make the elements of the first row to be all ones.