Consider the matrix, \(P=\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right)\)
Let the transpose of a matrix \(X\) be denoted by \(\mathrm{X}^T\). Then the number of \(3 \times 3\) invertible matrices Q with integer entries, such that \(Q^{-1}=Q^T\) and \(P Q=Q P\) is

\(\begin{aligned}
& \mathrm{PQ}=\mathrm{QP} \Rightarrow\left[\begin{array}{ccc}
2 \mathrm{a}_1 & 2 \mathrm{~b}_1 & 2 \mathrm{c}_1 \\
2 \mathrm{a}_2 & 2 \mathrm{~b}_2 & 2 \mathrm{c}_2 \\
3 \mathrm{a}_3 & 3 \mathrm{~b}_3 & 3 \mathrm{c}_3
\end{array}\right]=\left[\begin{array}{ccc}
2 \mathrm{a}_1 & 2 \mathrm{~b}_1 & 3 \mathrm{c}_1 \\
2 \mathrm{a}_2 & 2 \mathrm{~b}_2 & 3 \mathrm{c}_2 \\
2 \mathrm{a}_3 & 2 \mathrm{~b}_3 & 3 \mathrm{c}_3
\end{array}\right] \\
& \mathrm{c}_1=0, \mathrm{c}_2=0, \mathrm{a}_3=0, \mathrm{~b}_3=0 \\
& \mathrm{Q}=\left[\begin{array}{ccc}
\mathrm{a}_1 & \mathrm{~b}_1 & 0 \\
\mathrm{a}_2 & \mathrm{~b}_2 & 0 \\
0 & 0 & \mathrm{c}_3
\end{array}\right] \\
& \mathrm{a}_1 \mathrm{a}_2+\mathrm{b}_1 \mathrm{~b}_2=0 \\
& \mathrm{a}_1^2+\mathrm{b}_1^2=1, \mathrm{a}_2^2+\mathrm{b}_2^2=1, \mathrm{c}_3^2=1
\end{aligned}\)
\(\begin{array}{lllll}
a_1 & b_1 & a_2 & b_2 & c_3 \\
1 & 0 & 0 & 1,-1 & +1,-1 \\
-1 & 0 & 0 & 1,-1 & 1,-1 \\
0 & 1 & 1,-1 & 0 & 1,-1 \\
0 & -1 & 1,-1 & 0 & 1,-1
\end{array}\)
Total 16 matrices