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

$(a)\;2^{10}\qquad(b)\;2^{11}\qquad(c)\;2^{12}\qquad(d)\;2^{13}$

We have $\mid Q\mid=\begin{vmatrix}2^2a_{11}&2^3a_{12}&2^4a_{13}\\2^3a_{21}&2^4a_{22}&2^5a_{23}\\2^4a_{31}&2^5a_{32}&2^6a_{33}\end{vmatrix}$
$\Rightarrow 2^2.2^3.2^4\begin{vmatrix}a_{11}&a_{12}&a_{13}\\2a_{21}&2a_{22}&a_{23}\\2^2a_{31}&2^2a_{32}&2^2a_{33}\end{vmatrix}$
$\Rightarrow 2^9.2.2^2\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}$
$\Rightarrow 2^{12}\times \mid P\mid$
$\Rightarrow 2^{12}\times 2=2^{13}$
Hence (d) is the correct answer.