Let $ P = [p_{ij}] $ and $ Q = [q_{ij}] $ 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(\text{adj(adj } P)) $ is:

$ Q = \left[ \matrix{ 2p_{11} & 2^2 p_{12} & 2^3 p_{13} \cr 2^2 p_{21} & 2^3 p_{22} & 2^4 p_{23} \cr 2^3 p_{31} & 2^4 p_{32} & 2^5 p_{33} } \right] $

Factor out:

$ = 2^1 \cdot 2^2 \cdot 2^3 \times 2^2 \cdot 2^3 \cdot 2^4 \times 2^3 \cdot 2^4 \cdot 2^5 \cdot |P| $

$ = 2^9 |P| = 2^{10} $

$ \Rightarrow |P| = 2 $$ |\text{adj(adj } P)| = |P|^{(n-1)^2} = |P|^4 = 2^4 = 16 $