1.
Consider the matrix
P=(200020003).P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}.

Let the transpose of a matrix XX be denoted by XTX^T. Then the number of 3×33 \times 3 invertible matrices QQ with integer entries, such that
Q−1=QT and PQ=QP,Q^{-1} = Q^T \text{ and } PQ = QP,

is