Let A = \ a _ ij \ be a 3 × 3 matrix, where a _ ij = \ array ll (- 1) ^ j - i ifi j array ) then det(3 Adj (2 A ^ - 1 )) is equal to ________.

Correct answer: 0 We have, a _ ij = \ array ll (- 1) ^ j - i ifi j array ) The, A = [ cases 2 & - 1 & 1 - 1 & 2 & - 1 1 & - 1 & 2 cases ] ⇒ |A| = 4 Now, |3 adj (2 A ^ - 1 )| = 3 ^ 3 |adj (2 A ^ - 1 )| ∵ |k adj ( A ^ - 1 )| = k ^ n |adj ( A ^ - 1 )| ⇒ |3 adj (2 A ^ - 1 )| = 3 ^ 3 · 2 ^ 6 | A ^ - 1 | ^ 2 ∵ |adj (k A ^ - 1 )| = ( k ^ (n - 1) ) ^ n |adj ( A ^ - 1 )| = ( k ^ (n - 1) ) ^ n | A ^ - 1 | ^ n - 1 ⇒ |3 adj (2 A ^ - 1 )| = 12 ^ 3 | A ^ - 1 | ^ 2 = 12 ^ 3 | A | ^ 2 = 12 ^ 3 16 = 108

JEE Main 202120 JulyEvening Shift

Exam: JEE Main
Year: 2021
Date: 20 July
Shift: Evening Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2021 · 20 July · Evening Shift · Mathematics · Matrices

Let A = \ a _ ij \ be a 3 × 3 matrix, where a _ ij = \ array ll (-...

Let

A =

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

be a

3 \times 3

matrix, where

{a}_{ij} = \left\{

\begin{array}{ll}{\left(- 1\right)}^{j - i}ifi j\end{array}

\right)

then det\left(3 Adj \left(2{A}^{- 1}\right)\right)

is equal to ________.

Correct answer

Step-by-step solution

We have,

${a}_{ij} = \left\{$ $\begin{array}{ll}{\left(- 1\right)}^{j - i}ifi j\end{array}$ $\right)The,$ A = \left[ $\begin{cases}2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2\end{cases}$ \right] $$ ⇒ \left|A\right| = 4 $Now,\left|3 adj \left(2{A}^{- 1}\right)\right| = {3}^{3}\left|adj \left(2{A}^{- 1}\right)\right|$

$∵ \left|k adj \left({A}^{- 1}\right)\right| = {k}^{n}\left|adj \left({A}^{- 1}\right)\right|$

$⇒ \left|3 adj \left(2{A}^{- 1}\right)\right| = {3}^{3} \cdot {2}^{6}{\left|{A}^{- 1}\right|}^{2}$

$∵ \left|adj \left(k{A}^{- 1}\right)\right| = {\left({k}^{\left(n - 1\right)}\right)}^{n}\left|adj \left({A}^{- 1}\right)\right| = {\left({k}^{\left(n - 1\right)}\right)}^{n}{\left|{A}^{- 1}\right|}^{n - 1}$

⇒ \left|3 adj \left(2{A}^{- 1}\right)\right| = {12}^{3}{\left|{A}^{- 1}\right|}^{2} = \dfrac{{12}^{3}}{| A{|}^{2}} = \dfrac{{12}^{3}}{16} = 108

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