Let P = [aij] be a 3 × 3 matrix and let Q = [bij], where bij = 2i+jaij for 1 \leq i, j \leq 3. If the determinant of P is 2, then the determinant of the matrix Q is

Engineering

Mathematics

Algebra of Matrices

Question

Let P = [a_ij] be a 3 × 3 matrix and let Q = [b_ij], where b_ij = 2^i+ja_ij for 1 $\leq$ i, j $\leq$ 3. If the determinant of P is 2, then the determinant of the matrix Q is

2¹⁰

2¹³

2¹²

2¹¹

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$Let P = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}] \Rightarrow \det P = | \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$

$then \det Q = | \begin{matrix} 2^{2} a_{11} & 2^{3} a_{12} & 2^{4} a_{13} \\ 2^{3} a_{21} & 2^{4} a_{22} & 2^{5} a_{23} \\ 2^{4} a_{31} & 2^{5} a_{32} & 2^{6} a_{33} \end{matrix} | = 2^{2} 2^{3} 2^{4} [\begin{matrix} a_{11} & 2 a_{12} & 2^{2} a_{13} \\ a_{21} & 2 a_{22} & 2^{2} a_{23} \\ a_{31} & 2 a_{32} & 2^{2} a_{33} \end{matrix}]$

$= 2^{9} \cdot 2 \cdot 2^{2} = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}] = 2^{12} detP = 2^{13}$