Let A be a n × n matrix such that |A| = 2 . If the determinant of the matrix Adj (2 . Adj (2 A ^ - 1 )) is 2 ^ 84 , then n is equal to _____ .

Correct answer: 0 Given, |A| = 2 Now simplifying, |Adj(2 . Adj (2 A ^ - 1 ))| = |2 · Adj (2 A ^ - 1 )| ^ n - 1 = ( 2 ^ n |Adj (2 A ^ - 1 )|) ^ n - 1 = ( 2 ^ n |2 A ^ - 1 | ^ n - 1 ) ^ n - 1 = 2 ^ n(n - 1) ( ( 2 ^ n | A ^ - 1 |) ^ (n - 1) ) ^ (n - 1) ∴ | A ^ - 1 | = (1)/(|A|) = (1)/(2) = 2 ^ n(n - 1) ( ( 2 ^ (n - 1) ) ^ (n - 1) ) ^ (n - 1) = 2 ^ n(n - 1) + (n - 1) ^ 3 = 2 ^ 84 Now comparing both side we get, n(n - 1) + (n - 1) ^ 3 = 84 ⇒ (n - 1)(n + n ^ 2 - 2n + 1) = 84 ⇒ (n - 1)( n ^ 2 - n + 1) = 4 × 21 Now if n - 1 = 4 ⇒ n = 5 now checking n ^ 2 - 3n + 1 = 25 - 5 + 1 = 21 Hence, n = 5

JEE Main 202331 JanuaryEvening Shift

Exam: JEE Main
Year: 2023
Date: 31 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2023 · 31 January · Evening Shift · Mathematics · Matrices

Let A be a n × n matrix such that |A| = 2 . If the determinant of...

Let

A

be a

n \times n

matrix such that \left|A\right| = 2

. If the determinant of the matrix

Adj \left(2 . Adj

\left(2{A}^{- 1}\right)\right)

{2}^{84}

, then

n

is equal to _____ .

Correct answer

Step-by-step solution

Given,

$\left|A\right| = 2$

Now simplifying,

\left|Adj\left(2 . Adj \left(2{A}^{- 1}\right)\right)\right| $$ = {\left|2 \cdot Adj \left(2{A}^{- 1}\right)\right|}^{n - 1} $$ = {\left({2}^{n}\left|Adj \left(2{A}^{- 1}\right)\right|\right)}^{n - 1} $$ = {\left({2}^{n}{\left|2{A}^{- 1}\right|}^{n - 1}\right)}^{n - 1} $$ = {2}^{n\left(n - 1\right)}{\left({\left({2}^{n}\left|{A}^{- 1}\right|\right)}^{\left(n - 1\right)}\right)}^{\left(n - 1\right)} $$ ∴ \left|{A}^{- 1}\right| = \dfrac{1}{\left|A\right|} = \dfrac{1}{2} $$ = {2}^{n\left(n - 1\right)}{\left({\left({2}^{\left(n - 1\right)}\right)}^{\left(n - 1\right)}\right)}^{\left(n - 1\right)} $$ = {2}^{n\left(n - 1\right) + {\left(n - 1\right)}^{3}} = {2}^{84} $Now comparing both side we get,$ n\left(n - 1\right) + {\left(n - 1\right)}^{3} = 84 $$ ⇒ \left(n - 1\right)\left(n + {n}^{2} - 2n + 1\right) = 84 $$ ⇒ \left(n - 1\right)\left({n}^{2} - n + 1\right) = 4 \times 21 $Now if$ n - 1 = 4 ⇒ n = 5 $now checking$ {n}^{2} - 3n + 1 = 25 - 5 + 1 = 21 $Hence,$ n = 5$

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