[THOUGHT] Let me analyze this problem step by step.

We need to find 3×3 invertible matrices P with integer entries such that:

1. P^(-1) = P^T (P is orthogonal)
2. PM = MP (P commutes with M)

Since P^(-1) = P^T, we have P^T P = I, meaning P is an orthogonal matrix.

For P to have integer entries and be orthogonal, each row and column must have norm 1 and be orthogonal to other rows/columns. With integer entries, this means each row/column has exactly one ±1 and the rest are 0. So P is a signed permutation matrix.

Now for the condition PM = MP. Let's see what this means.

M = diag(2, 2, 3)

For PM = MP, if P = (p_ij), then:

• (PM)_ij = p_ij * M_jj (since M is diagonal)
• (MP)_ij = M_ii * p_ij

So we need p_ij * M_jj = M_ii * p_ij for all i, j.

This means p_ij ≠ 0 implies M_ii = M_jj.

Since M_11 = M_22 = 2 and M_33 = 3, we have:

• p_ij can be nonzero only if (i,j) ∈ {(1,1), (1,2), (2,1), (2,2), (3,3)}

So P must have the block structure: P = [[a, b, 0], [c, d, 0], [0, 0, e]]

where the 2×2 block [[a,b],[c,d]] is an orthogonal matrix with integer entries, and e = ±1.

For the 2×2 block to be orthogonal with integer entries, it must be a signed permutation matrix. The 2×2 orthogonal matrices with integer entries are:

• [[1,0],[0,1]], [[−1,0],[0,−1]], [[1,0],[0,−1]], [[−1,0],[0,1]]
• [[0,1],[1,0]], [[0,−1],[−1,0]], [[0,1],[−1,0]], [[0,−1],[1,0]]

That's 8 matrices for the 2×2 block, and 2 choices for e (±1).

So total = 8 × 2 = 16 matrices.

The answer asks for the square of this number: 16² = 256.

Let me verify this with Python. [/THOUGHT] [PYTHON] import numpy as np from itertools import product

Define M

M = np.array([[2, 0, 0], [0, 2, 0], [0, 0, 3]])

count = 0 valid_matrices = []

Generate all 3x3 matrices with entries in {-1, 0, 1}

for entries in product([-1, 0, 1], repeat=9): P = np.array(entries).reshape(3, 3)

# Check if P is invertible (det != 0)
det = np.linalg.det(P)
if abs(det) < 0.5:  # det should be ±1 for integer orthogonal matrices
    continue

# Check P^T P = I (orthogonal condition)
PTP = P.T @ P
if not np.allclose(PTP, np.eye(3)):
    continue

# Check PM = MP
PM = P @ M
MP = M @ P
if not np.allclose(PM, MP):
    continue

count += 1
valid_matrices.append(P)

print(f"Number of valid matrices: {count}") print(f"Square of the number: {count**2}")

Let's also print the matrices to verify

print("\nValid matrices:") for i, P in enumerate(valid_matrices): print(f"Matrix {i+1}:") print(P) print() [/PYTHON]

[THOUGHT] The Python code confirms that there are exactly 16 invertible 3×3 matrices P with integer entries satisfying both P^(-1) = P^T and PM = MP. The square of this number is 16² = 256. [/THOUGHT]

\boxed{256}