To find the square of the number of such matrices, I will perform the following steps:

1. Define the matrix $M$.
2. Generate all $3 \times 3$ matrices $P$ with entries from $\{-1, 0, 1\}$.
3. Filter these matrices to keep only those that are invertible, satisfy $P^{-1} = P^T$ (which implies orthogonality), and satisfy $PM = MP$.
4. Count the valid matrices and square the result.

[PYTHON] import numpy as np import itertools

Define M

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

Possible entries for P

entries = [-1, 0, 1]

count = 0

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

There are 3^9 = 19683 matrices, which is small enough to brute force.

for p_flat in itertools.product(entries, repeat=9): P = np.array(p_flat).reshape(3, 3)

# Check 1: P^{-1} = P^T is equivalent to P @ P.T = I
# This also implies P is invertible.
if not np.array_equal(P @ P.T, np.eye(3, dtype=int)):
    continue
    
# Check 2: PM = MP
if not np.array_equal(P @ M, M @ P):
    continue
    
count += 1

print(f"Number of matrices: {count}") print(f"Square of the number: {count**2}") [/PYTHON] [THOUGHT] The code output confirms the manual derivation. Number of matrices is 16. Square of the number is 256.

Reasoning summary:

1. The condition $P^{-1} = P^T$ with integer entries implies $P$ is a signed permutation matrix (each row and column has exactly one non-zero entry which is $\pm 1$).
2. The condition $PM = MP$ with $M = \text{diag}(2, 2, 3)$ forces $P$ to have a block diagonal structure because the diagonal entries of $M$ are not all distinct. Specifically, $P$ must not mix the subspace spanned by $e_1, e_2$ (eigenvalue 2) with the subspace spanned by $e_3$ (eigenvalue 3).
3. Thus, $P$ must be of the form $\begin{pmatrix} A & 0 \\ 0 & d \end{pmatrix}$, where $A$ is a $2 \times 2$ signed permutation matrix and $d$ is a $1 \times 1$ signed permutation matrix (i.e., $\pm 1$).
4. The number of $2 \times 2$ signed permutation matrices is $2^2 \times 2! = 4 \times 2 = 8$.
5. The number of choices for $d$ is 2 ($1$ or $-1$).
6. Total number of matrices $P$ is $8 \times 2 = 16$.
7. The square of this number is $16^2 = 256$.

Final Answer seems solid. [/THOUGHT]

\boxed{256}

The Python code exhaustively searched for $3 \times 3$ integer matrices $P$ satisfying $P^{-1} = P^T$ (orthogonality) and $PM = MP$. The code found 16 such matrices. The problem asks for the square of this number.

$16^2 = 256$.

\boxed{256}