Let $P=[a_{ij}]$ be a $3×3$ matrix and let $Q=[b_{ij}],$ where $b_{ij}=2^{i+j}a_{ij} $ for $1≤i, j ≤ 3.$ If the determinant of P is 2, then the determinant of the matrix Q, is

$2^{13}$

The correct answer is option (4) : $2^{13}$

We have, $Q=[b_{ij}],$ where $b_{ij}= 2^{i+j} a_{ij}$

$∴Q=\begin{bmatrix}b_{11} & b_{12} & b_{13}\\b_{21} & b_{22} & b_{23}\\b_{31} & b_{32} & b_{33}\end{bmatrix}$

$⇒Q=\begin{bmatrix}2^2a_{11} & 2^3a_{12} & 2^4a_{13}\\2^3a_{21} & 2^4a_{22} & 2^5a{23}\\2^4a_{31} & 2^5a_{32} & 2^6a_{33}\end{bmatrix}$

$⇒Q=2^2×2^3×2^4\begin{bmatrix}a_{11} & 2a_{12} & 2^2a_{13}\\2a_{21} & 2a_{22} & 2^2a{23}\\2a_{31} & 2a_{32} & 2^2a_{33}\end{bmatrix}$

$⇒Q=2^9×2×2^2\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a{23}\\a_{31} & a_{32} & a_{33}\end{bmatrix}=2^{12}P$

$⇒|Q|=2^{12}|P|=2^{12}×2=2^{13}$