Template:3-bit Walsh permutation/inverse nitpick

six edge cases of invertibility

To be precise, the group consists of the 168 binary matrices whose inverses are integer matrices, namely with entries 0, 1 and −1.
Not part of the group are the six matrices whose inverses do not contain integers:

binary

[[0, 1, 1],
 [1, 0, 1],
 [1, 1, 0]]

[[0, 1, 1],
 [1, 1, 0],
 [1, 0, 1]]

[[1, 0, 1],
 [0, 1, 1],
 [1, 1, 0]]

[[1, 0, 1],
 [1, 1, 0],
 [0, 1, 1]]

[[1, 1, 0],
 [0, 1, 1],
 [1, 0, 1]]

[[1, 1, 0],
 [1, 0, 1],
 [0, 1, 1]]

inverse

[[-0.5,  0.5,  0.5],
 [ 0.5, -0.5,  0.5],
 [ 0.5,  0.5, -0.5]]

[[-0.5,  0.5,  0.5],
 [ 0.5,  0.5, -0.5],
 [ 0.5, -0.5,  0.5]]

[[ 0.5, -0.5,  0.5],
 [-0.5,  0.5,  0.5],
 [ 0.5,  0.5, -0.5]]

[[ 0.5,  0.5, -0.5],
 [-0.5,  0.5,  0.5],
 [ 0.5, -0.5,  0.5]]

[[ 0.5, -0.5,  0.5],
 [ 0.5,  0.5, -0.5],
 [-0.5,  0.5,  0.5]]

[[ 0.5,  0.5, -0.5],
 [ 0.5, -0.5,  0.5],
 [-0.5,  0.5,  0.5]]

The following Python script shows that the 512 binary matrices consist of 338 singular matrices, 168 with integer inverses and 6 with non-integer inverses:

from itertools import product

def is_integer_matrix(matrix):
    for row in matrix:
        for entry in row:
            if entry % 1 > 0:
                return False
    return True

number_of_singular_matrices = 0
number_of_matrices_with_integer_inverses = 0
number_of_matrices_with_non_integer_inverses = 0

for m00, m01, m02, m10, m11, m12, m20, m21, m22 in product([0, 1], repeat=9):
    mat = np.array([[m00, m01, m02], [m10, m11, m12], [m20, m21, m22]])
    try:
        mat_inv = np.linalg.inv(mat)
        if is_integer_matrix(mat_inv):
            number_of_matrices_with_integer_inverses += 1
        else:
            number_of_matrices_with_non_integer_inverses += 1
    except np.linalg.LinAlgError:
        number_of_singular_matrices += 1
    
print('singular matrices:', number_of_singular_matrices)
print('matrices with integer inverses:', number_of_matrices_with_integer_inverses)
print('matrices with non-integer inverses:', number_of_matrices_with_non_integer_inverses)