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Determinants · Question 23

JEE Main 2025 (Online) 7th April Morning ShiftMCQMedium📈 Topic in 42% papers+4−1
1

Let A be a 3 3 matrix such that |adj(adj(adj A))| = 81. If S = \n Z : (|adj(adj A)| ) (n-1)22 = |A| (3n2 - 5n - 4 ) \, then n S |A (n2 + n ) | is equal to :

  1. 820
  2. 866
  3. 750
  4. 732

Answer

(4) 732

Solution

For a matrix of order 3: |adj A| = |A|2, |adj(adj A)| = |A|4, |adj(adj(adj A))| = |A|8

|A|8 = 81 = 34 |A|2 = 3

Condition for S: (|A|4 ) (n-1)22 = |A|3n2 - 5n - 4

|A|2(n-1)2 = |A|3n2 - 5n - 4

2n2 - 4n + 2 = 3n2 - 5n - 4

n2 - n - 6 = 0 n = 3 or n = -2

(both exponents are even: 8 and 18, so the sign of |A| does not matter) S = \-2, 3\

n = 3: |A12 | = |A|12 = 36 = 729

n = -2: |A2 | = |A|2 = 3

sum = 729 + 3 = 732

Answer: (4)

JEE Main 2025 (Online) 7th April Morning Shift

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