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

Let P = [a_ij] be a 3 × 3 matrix and let Q = [b_ij], where… — JEE Main 2012 Mathematics PYQ

AIEEE 2012

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

2

, then the determinant of the matrix

Q

A.
$2^{10}$
B.
$2^{11}$
C.
$2^{12}$
D.
$2^{13}$

Show answer and solution

Correct option: D

Solution

Q is formed by multiplying each entry a_ij by 2^(i+j). Factor out powers of 2 from each row and column: from row i, every entry gets a factor of 2ⁱ, and from column j, every entry gets a factor of 2^j. So det(Q) = (2¹ · 2² · 2³) · (2¹ · 2² · 2³) · det(P) = 2⁶ · 2⁶ · 2 = 2¹³.

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