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

$2^{13}$

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

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

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

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

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

$⇒|Q|=2^{12}|P|=2^{12}× 2=2^{13}$