Given P = [p_ij]_{3 × 3} and Q = [q_ij]_{3 × 3} are 3 × 3 matrices, where q_ij = 2^{i + j – 1} p_ij. If |Q| = 2¹⁰, then the value of |adj(adj(P))| is

The correct answer is 16.

Let us solve the problem step-by-step.
We are given two $3\times 3$ matrices, $P=\left[p_{ij}\right]$ and $Q=\left[q_{ij}\right]$, such that:
$q_{ij}=2^{i+j-1}{p}_{ij}$

Let us write the matrix $Q$ explicitly in terms of the elements of $P$:
$Q=\left[\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}\right]=\left[\begin{array}{ccc}{2}^1{p}_{11}& 2^2{p}_{12}& 2^3{p}_{13}\\ 2^2{p}_{21}& 2^3{p}_{22}& 2^4{p}_{23}\\ 2^3{p}_{31}& 2^4{p}_{32}& 2^5{p}_{33}\end{array}\right]$

Now, let us calculate the determinant of $Q$, denoted as $\left|Q\right|$:
$\left|Q\right|=\left|\begin{array}{ccc}{2}^1{p}_{11}& 2^2{p}_{12}& 2^3{p}_{13}\\ 2^2{p}_{21}& 2^3{p}_{22}& 2^4{p}_{23}\\ 2^3{p}_{31}& 2^4{p}_{32}& 2^5{p}_{33}\end{array}\right|$

We can factor out common terms from the rows and columns of this determinant:
From row 1 ($R_1$), take out a factor of $2^1$.
From row 2 ($R_2$), take out a factor of $2^2$.
From row 3 ($R_3$), take out a factor of $2^3$.
This gives:
$\left|Q\right|=2^1·2^2·2^3\left|\begin{array}{ccc}{p}_{11}& 2^1{p}_{12}& 2^2{p}_{13}\\ p_{21}& 2^1{p}_{22}& 2^2{p}_{23}\\ p_{31}& 2^1{p}_{32}& 2^2{p}_{33}\end{array}\right|$

Next, we can factor out common terms from the columns:
From column 2 ($C_2$), take out a factor of $2^1$.
From column 3 ($C_3$), take out a factor of $2^2$.
This gives:
$\left|Q\right|=(2^1·2^2·2^3)·(2^1·2^2)\left|\begin{array}{ccc}{p}_{11}& p_{12}& p_{13}\\ p_{21}& p_{22}& p_{23}\\ p_{31}& p_{32}& p_{33}\end{array}\right|$

Simplifying the powers of 2 and identifying the remaining determinant as $\left|P\right|$:
$\left|Q\right|=2^{1+2+3+1+2}·\left|P\right|$
$\left|Q\right|=2^9·\left|P\right|$

We are given that $\left|Q\right|=2^{10}$. Substituting this value:
$2^{10}=2^9·\left|P\right|$
Dividing both sides by $2^9$:
$\left|P\right|=2$

Now, we need to find the value of $\left|\text{adj}\right(\text{adj}\left(P\right)\left)\right|$.
For a matrix $A$ of order $n\times n$, we have the standard property:
$\left|\text{adj}\right(\text{adj}\left(A\right)\left)\right|={\left|A\right|}^{{(n-1)}^2}$

Here, $P$ is a $3\times 3$ matrix, so $n=3$.
Substituting $n=3$ and $\left|P\right|=2$:
$\left|\text{adj}\right(\text{adj}\left(P\right)\left)\right|={\left|P\right|}^{{(3-1)}^2}=2^{2^2}=2^4=16$

Thus, the value of $\left|\text{adj}\right(\text{adj}\left(P\right)\left)\right|$ is 16.