Let $P=\left[a_{i j}\right]$ be $3 \times 3$ matrix and let $Q=\left[b_{i j}\right]$ be a $3 \times 3$ matrix, where $b_{i j}=2^{i+j} a_{i j}$ for $1 \leq i, j \leq 3$. If the determinant of $P$ is 2 , than the determinant of the matrix $Q$ is

$2^{13}$

$P=\left[a_{i j}\right] \quad \theta=\left[b_{i j}\right]$

$b_{i j}=2^{i+j} a_{1 j}$

$P=\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right] \quad Q=\left[\begin{array}{lll}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33}
\end{array}\right]$

substituting values of $b_{i j}$

$Q=\left[\begin{array}{ccc}2^2 b_{11} & 2^3 b_{12} & 2^4 b_{13} \\ 2^3 b_{21} & 2^4 b_{22} & 2^5 b_{23} \\ 2^4 b_{31} & 2^5 b_{32} & 2^6 b_{33}\end{array}\right]$

as  $|P|=2$

$|Q|=\left|\begin{array}{ccc}2^2 b_{11} & 2^3 b_{12} & 2^4 b_{13} \\ 2^3 b_{21} & 2^4 b_{22} & 2^5 b_{23} \\ 2^4 b_{31} & 2^5 b_{32} & 2^6 b_{33}\end{array}\right|$

Taking 2's power out from each column 

$|Q|=2^2 2^3 2^4\left|\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ 2 a_{21} & 2 a_{22} & 2 a_{23} \\ 2 a_{31}^2 & 2 a_{32}^2 & 2^2 a_{33}\end{array}\right| \Rightarrow$ Taking out 2 from R₂ and 2² from R₃

$|Q|=2^2 2^3 2^4 \times 2 \times 2^2 = \underbrace{\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|}_{|P|=2}$

$|Q| = 2^{2+3+4+2+1}×2$

$=2^{12}×2=2^{13}=|Q|$