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MATL, (削除) 9 (削除ここまで) 8 bytes

tY*zUGn=

Inputs a matrix. Gives 1 or 0 for Hadamard or non-Hadamard, respectively.

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How it works

t % Implicit input: matrix M, of size n, with 1 or −1 entries. Duplicate
Y* % Matrix product, M*M. This is a scalar matrix if and only if M is Hadamard
z % Number of non-zeros. Gives n if and only if M is Hadamard
U % Square 
G % Push input again
n % Number of entries. Gives n^2
= % Equal? Implicit display