[THOUGHT] I will brute-force over all 3x3 integer matrices with entries in {-1,0,1}, check which are orthogonal (P^T P = I), invertible (det ±1), and commute with M (PM = MP). Then count them and square the count. This is feasible since there are only 3^9 = 19683 matrices in that range, and any integer orthogonal matrix must have entries in {-1,0,1} anyway. [/THOUGHT] [PYTHON] import itertools import numpy as np

M = np.diag([2,2,3])

def is_orthogonal_int(P): PT = P.T prod = PT @ P return np.array_equal(prod, np.eye(3, dtype=int))

def commutes_with_M(P): return np.array_equal(P @ M, M @ P)

count = 0 valid_matrices = []

vals = [-1,0,1] for entries in itertools.product(vals, repeat=9): P = np.array(entries, dtype=int).reshape(3,3) # Quick determinant check for invertibility det = round(np.linalg.det(P)) if det not in (1, -1): continue if not is_orthogonal_int(P): continue if not commutes_with_M(P): continue count += 1 valid_matrices.append(P)

print("Number of valid matrices P:", count) print("Square of this number:", count**2)

Optionally inspect distinct matrices

for P in valid_matrices:

print(P, "\n")

[/PYTHON]

[THOUGHT] Using the verified Python enumeration of all integer orthogonal 3×3 matrices commuting with M, there are 16 such matrices P. The problem asks for the square of this number, which is 256. \boxed{256}