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.

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.

Edit: As Gerry mentions, there are examples of smaller order that are not very hard to find. (It helps to start with an integral matrix with determinant divisible by a power of an integer.)