They are not equivalent.  Take a (15,8,8) design ( or the 15 by 15 0-1 matrix associated with the order 16 Sylvester-Hadamard matrix ). Multiply all elements by 1/2, and call the result A.  AA is an integer matrix equal to 2I + 2J, and so every larger power of A is also an integer matrix. But A is half (!?!) of an integer matrix with nonnegative entries and determinant of size 2.