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

JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Determinants

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

Answer: (D)

∣ Q ∣ = 2^{2} a_{11} 2^{3} a_{21} 2^{4} a_{31} 2^{3} a_{12} 2^{4} a_{22} 2^{5} a_{32} 2^{4} a_{13} 2^{5} a_{23} 2^{6} a_{33}

. Factor out

2^{2}, 2^{3}, 2^{4}

from rows 1, 2, 3 respectively:

∣ Q ∣ = 2^{2 + 3 + 4} a_{11} a_{21} a_{31} 2 a_{12} 2 a_{22} 2 a_{32} 2^{2} a_{13} 2^{2} a_{23} 2^{2} a_{33}

. Factor out

2, 2^{2}

from columns 2, 3:

∣ Q ∣ = 2^{9} \cdot 2 \cdot 2^{2} ∣ P ∣ = 2^{12} \cdot 2 = 2^{13}

JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Single correct option