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Let $P=\left[a_{i j}\right]$ be a $3 \times 3$ matrix and let $Q=\left[b_{i j}\right]$, where $b_{i j}=2^{i+j} a_{i j}$ for $1 \leq i, j \leq 3$. If the determinant of $P$ is 2 , then the determinant of the matrix $Q$ is
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The correct answer is:
$2^{13}$
$|Q|=\left|\begin{array}{lll}2^{2} a_{11} & 2^{3} a_{12} & 2^{4} a_{13} \\ 2^{3} a_{21} & 2^{4} a_{22} & 2^{5} a_{23} \\ 2^{4} a_{31} & 2^{5} a_{32} & 2^{6} a_{33}\end{array}\right|$
$=2^{2} \cdot 2^{3} \cdot 2^{4}\left|\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ 2 a_{21} & 2 a_{22} & 2 a_{23} \\ 2^{2} a_{31} & 2^{2} a_{32} & 2^{2} a_{33}\end{array}\right|$
$=2^{9} \cdot 2 \cdot 2^{2}\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$
$=2^{12} \times|P|=2^{12} \times 2=2^{13}$
$=2^{2} \cdot 2^{3} \cdot 2^{4}\left|\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ 2 a_{21} & 2 a_{22} & 2 a_{23} \\ 2^{2} a_{31} & 2^{2} a_{32} & 2^{2} a_{33}\end{array}\right|$
$=2^{9} \cdot 2 \cdot 2^{2}\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$
$=2^{12} \times|P|=2^{12} \times 2=2^{13}$
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