It is a necessary condition for a Hadamard matrix of dimension n to exist that n = 1, 2 or a multiple of 4.
To see this, assume that H is a Hadamard matrix of dimension n > 2. without loss of generality, we may first multiply columns so as to obtain 1 in each entry of the top row, and next permute the columns so as to obtain the following picture for the first three rows.
| ++ ··· + | ++ ··· + | ++ ··· + | ++ ··· + |
| ++ ··· + | ++ ··· + | -- ··· - | -- ··· - |
| ++ ··· + | -- ··· - | ++ ··· + | -- ··· - |
| a columns | b columns | c columns | d columns |
Here, + represents 1 and - represents -1. The condition HHT = 1 implies that the inner products of rows are zero. Computing these for the inner products of the first two rows (possibly two equal rows), we find:
| a + b + c + d | = n |
| a + b - c - d | = 0 |
| a - b + c - d | = 0 |
| a - b - c + d | = 0 |
Straightforward solving gives
n = 4a, whence n = 0 mod 4.