Question 4

The condition $PQ = QP$ implies that $Q$ must commute with the diagonal matrix $P$.

Since $P = \text{diag}(2, 2, 3)$, the matrix $Q$ must be block-diagonal of the form: $Q = \begin{pmatrix} A & 0 \\ 0 & d \end{pmatrix}$ where $A$ is a $2 \times 2$ matrix and $d$ is a $1 \times 1$ matrix.

The condition $Q^{-1} = Q^T$ means $Q$ is an orthogonal matrix ($Q^T Q = I$).

For $Q$ to have integer entries and be orthogonal, each row and column must be a unit vector with integer components. This means each row/column must contain exactly one entry from $\{1, -1\}$ and the rest must be $0$.

1. For the $1 \times 1$ block $d$, $d^2 = 1 \implies d \in \{1, -1\}$ (2 choices).

2. For the $2 \times 2$ block $A$, it must be an integer orthogonal matrix. The possible forms are permutations of $\text{diag}(\pm 1, \pm 1)$:

• $\begin{pmatrix} \pm 1 & 0 \\ 0 & \pm 1 \end{pmatrix}$ (4 matrices)
• $\begin{pmatrix} 0 & \pm 1 \\ \pm 1 & 0 \end{pmatrix}$ (4 matrices) This gives a total of $4 + 4 = 8$ choices for $A$.

The total number of matrices $Q = 8 \times 2 = 16$.