Let $P = \left\lbrack p_{ij} \right\rbrack$ and $Q = \left\lbrack q_{ij} \right\rbrack$ be two squar

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Let $P = \left\lbrack p_{ij} \right\rbrack$ and $Q = \left\lbrack q_{ij} \right\rbrack$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)}p_{ij}$ and $det(Q) = 2^{10}$. Then the value of $det(adj(adjP))$ is :

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$\left| \begin{matrix} 2p_{11} & 2^{2}p_{12} & 2^{3}p_{13} \\ 2^{2}p_{21} & 2^{3}p_{22} & 2^{4}p_{23} \\ 2^{3}p_{31} & 2^{4}p_{32} & 2^{5}p_{33} \end{matrix} \right| = 2^{10}$

$${2^{2} \cdot 2 \cdot 2^{3}\left| \begin{matrix} p_{11} & p_{12} & p_{13} \\ 2p_{21} & 2p_{22} & 2p_{23} \\ 2^{2}p_{31} & 2^{2}p_{32} & 2^{2}p_{33} \end{matrix} \right| = 2^{10} }{2^{9}\left| \begin{matrix} p_{11} & p_{12} & p_{13} \\ p_{21} & p_{22} & p_{23} \\ p_{31} & p_{32} & p_{33} \end{matrix} \right| = 2^{10} \Rightarrow |P| = 2 }{|adj(adj(P))| = |P|^{(n - 1)^{2}} = |P|^{4} = 2^{4} = 16}$$

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