import numpy as np

def generate_orthogonal_matrix(n, id_num):
    """Deterministic orthogonal matrix for given ID and dimension."""
    rng = np.random.RandomState(id_num * 17 + 42)
    A = rng.randn(n, n)
    Q, _ = np.linalg.qr(A)
    if np.linalg.det(Q) < 0:
        Q[:, 0] = -Q[:, 0]
    return Q