Question

Consider the matrix $$ P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. $$ Let the transpose of a matrix $ X $ be denoted by $ X^T $. Then the number of $ 3 \times 3 $ invertible matrices $ Q $ with integer entries, such that $$ Q^{-1} = Q^T \quad \text{and} \quad PQ = QP, $$ is:

• A. 32
• B. 8
• C. 16
• D. 24

Hint:

Integer orthogonal matrices form a finite group and are always signed permutation matrices. Use eigenspace preservation to restrict their form when commuting with a diagonal matrix.

Answer 1

The Correct Option is B

Step 1: Interpret the conditions We are told: - \( Q^{-1} = Q^T \) $\Rightarrow $ \( Q \) is an orthogonal matrix
. - \( Q \) is invertible with integer entries $\Rightarrow$ entries must be from \(−1, 0, 1\).
- \( PQ = QP \) $\Rightarrow$ \( Q \) commutes with the diagonal matrix \( P \).
Step 2: Structure of \( Q \) The matrix \( P \) has eigenvalues 2, 2, and 3. This means the eigenspace corresponding to 2 is 2-dimensional (spanned by standard basis vectors \( e_1, e_2 \)), and the eigenspace corresponding to 3 is 1-dimensional (spanned by \( e_3 \)).
Since \( Q \) must commute with \( P \), it must preserve these eigenspaces. So \( Q \) must be of the form: \[ Q = \begin{pmatrix} Q_1 & 0 
0 & \pm 1 \end{pmatrix} \] where \( Q_1 \) is a \( 2 \times 2 \) orthogonal matrix with integer entries. 
Step 3: Count such matrices The number of \( 2 \times 2 \) orthogonal matrices over integers is 4 (rotations and reflections in 2D with integer entries): - Identity, swap rows, sign flips, etc. And for the bottom-right corner (\( \pm1 \)), we have 2 choices. So total number of such matrices: \[ 4 \times 2 = \boxed{8} \]