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(D) Given $P = [a_{ij}]_{3 	imes 3}$ and $b_{ij} = 2^{i+j} a_{ij}$.$Q = [b_{ij}]_{3 	imes 3} = \begin{bmatrix} 2^{1+1}a_{11} & 2^{1+2}a_{12} & 2^{1+

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$2^{13}$

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(D) Given $P = [a_{ij}]_{3 	imes 3}$ and $b_{ij} = 2^{i+j} a_{ij}$.$Q = [b_{ij}]_{3 	imes 3} = \begin{bmatrix} 2^{1+1}a_{11} & 2^{1+2}a_{12} & 2^{1+3}a_{13} \ 2^{2+1}a_{21} & 2^{2+2}a_{22} & 2^{2+3}a_{23} \ 2^{3+1}a_{31} & 2^{3+2}a_{32} & 2^{3+3}a_{33} \end{bmatrix} = \begin{bmatrix} 4a_{11} & 8a_{12} & 16a_{13} \ 8a_{21} & 16a_{22} & 32a_{23} \ 16a_{31} & 32a_{32} & 64a_{33} \end{bmatrix}$.Taking common factors from each row:Row $1$ has a factor of $4 = 2^2$.Row $2$ has a factor of $8 = 2^3$.Row $3$ has a factor of $16 = 2^4$.$|Q| = (2^2 \cdot 2^3 \cdot 2^4) \begin{vmatrix} a_{11} & 2a_{12} & 4a_{13} \ a_{21} & 2a_{22} & 4a_{23} \ a_{31} & 2a_{32} & 4a_{33} \end{vmatrix}$.Taking common factors from each column:Column $2$ has a factor of $2 = 2^1$.Column $3$ has a factor of $4 = 2^2$.$|Q| = (2^2 \cdot 2^3 \cdot 2^4) \cdot (2^1 \cdot 2^2) \begin{vmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{vmatrix}$.$|Q| = 2^{2+3+4+1+2} |P| = 2^{12} \cdot 2 = 2^{13}$.

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:

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