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

Correct answer: D. 2^ 13 |Q|=| 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 array | =2^ 2 · 2^ 3 · 2^ 4 | 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 array | =2^ 9 · 2 · 2^ 2 | array lll a_ 11 & a_ 12 & a_ 13 a_ 21 & a_ 22 & a_ 23 a_ 31 & a_ 32 & a_ 33 array | =2^ 12 ×|P|=2^ 12 × 2=2^ 13

JEE Advanced 2012Paper 1

Exam: JEE Advanced
Year: 2012
Paper: Paper 1
Subject: Mathematics
Chapter: Determinants

JEE Advanced 2012 · Paper 1 · Mathematics · Determinants

Let P=[a_ i j ] be a 3 × 3 matrix and let Q=[b_ i j ] , where b_ i...

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

is 2 , then the determinant of the matrix

Q$ is

Options

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

Correct answer

D. $2^{13}$

Step-by-step solution

|Q|=\left|

\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}

\right|

=2^{2} \cdot 2^{3} \cdot 2^{4}\left|

\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}

\right|

=2^{9} \cdot 2 \cdot 2^{2}\left|

\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}

\right|

=2^{12} \times|P|=2^{12} \times 2=2^{13}

Practice Determinants on Quantrex Academy →

More from Determinants

If the system of linear equations: x+y+z=6 , x+2y+5z=10 , 2x+3y+λ z=μ has infinitely many… JEE Main 2026 The sum of all possible values of θ ∈ [0, 2π] , for which the system of equations : x 3θ… JEE Main 2026 If f: N arrow Z is defined by f(n) = vmatrix n & -1 & -5 -2n^2 & 3(2k+1) & 2k+1 -3n^3 &… JEE Main 2026 Consider the system of linear equations in x, y, z : x + 2y + tz = 0 , 6x + y + 5tz = 0… JEE Main 2026 If the system of equations: x+y+z=5 x+2y+3z=9 x+3y+λ z=μ has infinitely many solutions… JEE Main 2026 If the system of equations x + 5y + 6z = 4 , 2x + 3y + 4z = 7 , x + 6y + az = b has… JEE Main 2026 Let α, β ∈ ℝ be such that the system of linear equations x + 2y + z = 5 2x + y + α z = 5… JEE Main 2026 The system of linear equations x+y+z=6 2 x+5 y+a z=36 x+2 y+3 z=b has JEE Main 2026 Among the statements : I: If | array ccc 1 & α & β α & 1 & γ β & γ & 1 array |=| array… JEE Main 2026 Let n be the number obtained on rolling a fair die. If the probability that the system… JEE Main 2026 If the system of equations 3 x+y+4 z=3 2 x+α y-z=-3 x+2 y+z=4 has no solution, then the… JEE Main 2026 Let the system of equations aligned & x+5 y-z=1 & 4 x+3 y-3 z=7 & 24 x+y+λ z=μ aligned λ… JEE Main 2025 Let the system of equations : aligned & 2 x+3 y+5 z=9 & 7 x+3 y-2 z=8 & 12 x+3 y-(4+λ)… JEE Main 2025 If y(x)=| array ccc x & x & x+ x+1 27 & 28 & 27 1 & 1 & 1 array |, x ∈ ℝ , then (d^2… JEE Main 2025 If the system of equation aligned & 2 x+λ y+3 z=5 & 3 x+2 y-z=7 & 4 x+5 y+μ z=9 aligned… JEE Main 2025 If the system of linear equations aligned & 3 x+y+β z=3 & 2 x+α y-z=-3 & x+2 y+z=4… JEE Main 2025 Full Determinants list Determinants JEE Maths PYQs JEE Mathematics PYQs All JEE Advanced PYQs