Consider the matrix, P= ( array lll 2 & 0 & 0 0 & 2 & 0 0 & 0 & 3 array ) Let the transpose of a matrix X be denoted by X ^T . Then the number of 3 3 invertible matrices Q with integer entries, such that Q⁻¹=Q^T and P Q=Q P is

Step-by-step solution

( aligned & PQ = QP [ array ccc 2 a ₁ & 2 ~b ₁ & 2 c ₁ 2 a ₂ & 2 ~b ₂ & 2 c ₂ 3 a ₃ & 3 ~b ₃ & 3 c ₃ array ]= [ array ccc 2 a ₁ & 2 ~b ₁ & 3 c ₁ 2 a ₂ & 2 ~b ₂ & 3 c ₂ 2 a ₃ & 2 ~b ₃ & 3 c ₃ array ] & c ₁=0, c ₂=0, a ₃=0, ~b ₃=0 & Q = [ array ccc a ₁ & ~b ₁ & 0 a ₂ & ~b ₂ & 0 0 & 0 & c ₃ array ] & a ₁ a ₂+ b ₁ ~b ₂=0 & a ₁^2+ b ₁^2=1, a ₂^2+ b ₂^2=1, c ₃^2=1 aligned ) ( array lllll a₁ & b₁ & a₂ & b₂ & c₃ 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 array ) Total