Consider the matrix P = begin{bmatrix} 2 & 0 | vedclass

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(C) $Q^{-1} = Q^T \implies QQ^T = I$. Thus,$Q$ is an orthogonal matrix.Let $Q = \begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3

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$16$

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(C) $Q^{-1} = Q^T \implies QQ^T = I$. Thus,$Q$ is an orthogonal matrix.Let $Q = \begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix}$.The condition $PQ = QP$ implies:$\begin{bmatrix} 2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix} \begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix} = \begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix} \begin{bmatrix} 2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix}$$\begin{bmatrix} 2a_1 & 2b_1 & 2c_1 \ 2a_2 & 2b_2 & 2c_2 \ 3a_3 & 3b_3 & 3c_3 \end{bmatrix} = \begin{bmatrix} 2a_1 & 2b_1 & 3c_1 \ 2a_2 & 2b_2 & 3c_2 \ 2a_3 & 2b_3 & 3c_3 \end{bmatrix}$Comparing elements,we get $2c_1 = 3c_1 \implies c_1 = 0$,$2c_2 = 3c_2 \implies c_2 = 0$,$3a_3 = 2a_3 \implies a_3 = 0$,and $3b_3 = 2b_3 \implies b_3 = 0$.Since $Q$ is orthogonal,$Q^T Q = I$. For $Q = \begin{bmatrix} a_1 & b_1 & 0 \ a_2 & b_2 & 0 \ 0 & 0 & c_3 \end{bmatrix}$,the condition $Q^T Q = I$ implies $a_1^2 + a_2^2 = 1$,$b_1^2 + b_2^2 = 1$,$a_1b_1 + a_2b_2 = 0$,and $c_3^2 = 1$.Since entries are integers,$c_3 \in \{1, -1\}$. For the $2 	imes 2$ block,the possible orthogonal matrices with integer entries are $\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix}$.There are $8$ such $2 	imes 2$ matrices and $2$ choices for $c_3$,giving $8 	imes 2 = 16$ total matrices.

Consider the matrix $P = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$. 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 $Q^{-1} = Q^T$ and $PQ = QP$ is

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