Write out $Q$:

$Q=\left[\begin{array}{c}2p_{11}{2}^2{p}_{12}{2}^3{p}_{13}\\ 2^2{p}_{21}{2}^3{p}_{22}{2}^4{p}_{23}\\ 2^3{p}_{31}{2}^4{p}_{32}{2}^5{p}_{33}\end{array}\right]$

Pull factors out of rows: row 1 has factor $2$, row 2 has factor $2^2$, row 3 has factor $2^3$, giving $2^{1+2+3}=2^6$. From the remaining matrix, pull factors out of columns: column 1 factor $2^0$, column 2 factor $2$, column 3 factor $2^2$, giving $2^3$.

$\det (Q)=2^6\cdot 2^3\cdot |P|=2^9|P|=2^{10}\Rightarrow |P|=2$.

For an $n\times n$ matrix, $|\text{adj}(\text{adj}P)|=|P{|}^{(n-1{)}^2}=|P{|}^4$ (here $n=3$).

$\therefore |\text{adj}(\text{adj}P)|=2^4=16$.