Consider the matrixLet the transpose of a matrix be denoted by . Find the square of the number of 3 $$\times$$ 3 invertible matrices with integer entries, such that\
[THOUGHT] To solve this problem, we need to find the number of 3x3 invertible matrices ( P ) with integer entries that satisfy two conditions:
- ( P^{-1} = P^T ) (i.e., ( P ) is orthogonal).
- ( PM = MP ) (i.e., ( P ) commutes with the given matrix ( M )).
First, let's analyze the matrix ( M ). It is a diagonal matrix with entries 2, 2, and 3. The condition ( PM = MP ) implies that ( P ) must commute with ( M ). For a diagonal matrix ( M ) with distinct eigenvalues, the matrices ( P ) that commute with ( M ) are diagonal matrices. However, ( M ) has repeated eigenvalues (two 2's), so ( P ) can have non-diagonal entries in the positions corresponding to the repeated eigenvalues.
Next, we consider the orthogonality condition ( P^{-1} = P^T ). For integer matrices, this implies that ( P ) must be a signed permutation matrix (since orthogonal matrices with integer entries are signed permutation matrices). However, the condition ( PM = MP ) further restricts ( P ) to be a diagonal matrix with entries ( \pm 1 ) (since ( M ) has distinct eigenvalues except for the repeated 2's).
Given the structure of ( M ), the matrices ( P ) that satisfy both conditions must be diagonal matrices with entries ( \pm 1 ) in the diagonal, except possibly for the entries corresponding to the repeated eigenvalues (the first two diagonal entries of ( M )). However, the orthogonality condition restricts the entries to ( \pm 1 ).
Thus, the problem reduces to counting the number of diagonal matrices with entries ( \pm 1 ) on the diagonal that commute with ( M ). Since ( M ) is diagonal, any diagonal matrix ( P ) will commute with ( M ). Therefore, we need to count the number of diagonal matrices with entries ( \pm 1 ) on the diagonal.
For a 3x3 matrix, there are ( 2^3 = 8 ) possible diagonal matrices with entries ( \pm 1 ). However, we must ensure that ( P ) is invertible, which is always true for these matrices since their determinant is ( \pm 1 \neq 0 ).
Finally, we need to square the number of such matrices. So, the answer is ( 8^2 = 64 ).
[/THOUGHT] [PYTHON]
Verification: Count the number of diagonal matrices with entries ±1
count = 2 ** 3 # 2 choices (±1) for each of the 3 diagonal entries squared_count = count ** 2 print(squared_count) [/PYTHON]
[THOUGHT] The Python output confirms that the number of valid matrices ( P ) is 8, and squaring this gives 64. This matches our earlier reasoning. [/THOUGHT] \boxed{64}