Let $P = [a_{ij}]$ be a 3 x 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

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2¹¹

2¹²

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answer is D.

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Detailed Solution

$\begin{matrix} | Q | = |\begin{array}{lll} 2^{2} a_{11} & 2^{3} a_{12} & 2^{4} a_{13} \\ 2^{3} a_{21} & 2^{4} a_{22} & 2^{5} a_{23} \\ 2^{4} a_{31} & 2^{5} a_{32} & 2^{6} a_{33} \end{array}| \\ = 2^{2} \cdot 2^{3} \cdot 2^{4} |\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ 2 a_{21} & 2 a_{22} & 2 a_{23} \\ 2^{2} a_{31} & 2^{2} a_{32} & 2^{2} a_{33} \end{array}| \\ = 2^{9} \cdot 2 \cdot 2^{2} |\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}| \\ = 2^{12} | P | \\ = 2^{13} \end{matrix}$