Let A = [ a _ ij ] and B = [ b _ ij ] be two 3 × 3 real matrices such that b _ ij = (3) ^ (i + j - 2) a _ ij , where i, j = 1,2, 3 . If the determinant of B is 81 , then determinant of A i s

Correct answer: D. (1)/(9) |B| = | cases b _ 11 & b _ 12 & b _ 13 b _ 21 & b _ 22 & b _ 23 b _ 31 & b _ 32 & b _ 33 cases | = | cases 3 ^ 0 a _ 11 & 3 ^ 1 a _ 12 & 3 ^ 2 a _ 13 3 ^ 1 a _ 21 & 3 ^ 2 a _ 22 & 3 ^ 3 a _ 23 3 ^ 2 a _ 31 & 3 ^ 3 a _ 32 & 3 ^ 4 a _ 33 cases | ⇒ 81 = 3 ^ 3 ∙ 3 ∙ 3 ^ 2 |A| ⇒ 3 ^ 4 = 3 ^ 6 |A| ⇒ |A| = (1)/(9)

JEE Main 20207 JanuaryEvening Shift

Exam: JEE Main
Year: 2020
Date: 7 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Determinants

JEE Main 2020 · 7 January · Evening Shift · Mathematics · Determinants

Let A = [ a _ ij ] and B = [ b _ ij ] be two 3 × 3 real matrices...

Let

A =

\left[{a}_{ij}\right]

<em>and</em>

B =

\left[{b}_{ij}\right]

be two

3 \times 3

real matrices such that

{b}_{ij} = {\left(3\right)}

\left(i + j - 2\right)}{a}_{ij}

, where

i, j = 1,2, 3

. If the determinant of

B

81

, then determinant of

A

Options

A $\dfrac{1}{3}$
B $3$
C $\dfrac{1}{81}$
D $\dfrac{1}{9}$

Correct answer

D. $\dfrac{1}{9}$

Step-by-step solution

$\left|B\right| = \left|$ \begin{cases}{b}_{11} & {b}_{12} & {b}_{13} \\ {b}_{21} & {b}_{22} & {b}_{23} \\ {b}_{31} & {b}_{32} & {b}_{33}\end{cases} $\right| = \left|$ \begin{cases}{3}^{0}{a}_{11} & {3}^{1}{a}_{12} & {3}^{2}{a}_{13} \\ {3}^{1}{a}_{21} & {3}^{2}{a}_{22} & {3}^{3}{a}_{23} \\ {3}^{2}{a}_{31} & {3}^{3}{a}_{32} & {3}^{4}{a}_{33}\end{cases} $\right|$

$⇒ 81 = {3}^{3} ∙ 3 ∙ {3}^{2}\left|A\right| ⇒ {3}^{4} = {3}^{6}\left|A\right| ⇒ \left|A\right| = \dfrac{1}{9}$

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