Problem Analysis #12

[THOUGHT] We need integer invertible 3x3 matrices P with P^{-1}=P^T, so P is an integer orthogonal matrix; such matrices are exactly signed permutation matrices. The commutation condition PM=MP with M=diag(2,2,3) forces P to preserve the eigenspaces: it may permute/sign-change the first two coordinates (same eigenvalue 2) but cannot mix them with the third coordinate (eigenvalue 3). So P has block form diag(A, e) where A is a 2x2 signed permutation matrix (8 choices) and e is ±1 (2 choices), total 16. We verify by brute force enumeration of signed permutation matrices and count those commuting with M. [/THOUGHT] [PYTHON] import itertools import numpy as np

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

def signed_permutation_matrices(n=3): mats = [] for perm in itertools.permutations(range(n)): for signs in itertools.product([-1,1], repeat=n): P = np.zeros((n,n), dtype=int) for i,j in enumerate(perm): P[i,j] = signs[i] mats.append(P) return mats

count = 0 good = [] for P in signed_permutation_matrices(3): # orthogonality check: P^T P = I if not np.array_equal(P.T @ P, np.eye(3, dtype=int)): continue # commuting check if np.array_equal(P @ M, M @ P): count += 1 good.append(P)

print(count) print(count**2) [/PYTHON]

[THOUGHT] The Python computation shows there are 16 such matrices, so the requested square is $16^2=256$. $\boxed{256}$