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Determinants

JEE Main 2026 (Online) 22nd January Morning ShiftNumericalHard📈 Topic in 38% papers+4
Ques. 1JEE Main 2026, 22 January (Morning Shift)

Let A be a 3 3 matrix such that A + AT = O. If A bmatrix 1 \ -1 \ 0 bmatrix = bmatrix 3 \ 3 \ 2 bmatrix, A2 bmatrix 1 \ -1 \ 0 bmatrix = bmatrix -3 \ 19 \ -24 bmatrix and (adj(2 adj(A + I))) = (2) (3) (11) , , , are non-negative integers, then + + is equal to

Answer

18

Solution

A is skew-symmetric, so let A = bmatrix 0 & p & q \ -p & 0 & r \ -q & -r & 0 bmatrix. Then

A bmatrix 1 \ -1 \ 0 bmatrix = bmatrix -p \ -p \ r - q bmatrix = bmatrix 3 \ 3 \ 2 bmatrix

p = -3, r - q = 2

Next,

A2 bmatrix 1 \ -1 \ 0 bmatrix = A bmatrix 3 \ 3 \ 2 bmatrix = bmatrix 3p + 2q \ -3p + 2r \ -3q - 3r bmatrix = bmatrix -3 \ 19 \ -24 bmatrix

which gives -9 + 2q = -3, so q = 3, and 9 + 2r = 19, so r = 5 (check: -3(3 + 5) = -24). Hence

A + I = bmatrix 1 & -3 & 3 \ 3 & 1 & 5 \ -3 & -5 & 1 bmatrix

|A + I| = 1(1 + 25) + 3(3 + 15) + 3(-15 + 3) = 26 + 54 - 36 = 44

For a matrix M of order 3, |adj M| = |M|2 and |kM| = k3|M|. Let B = 2 adj(A + I); then |B| = 23 442, so

|adj B| = |B|2 = 26 444 = 26 28 114 = 214 30 114

Thus = 14, = 0 and = 4.

+ + = 18

Answer: 18

Keep practising — Determinants

Ques. 2JEE Main 2026, 22 January (Evening Shift)

Let n be the number obtained on rolling a fair die. If the probability that the system x - ny + z = 6 x + (n - 2)y + (n + 1)z = 8 (n - 1)y + z = 1 has a unique solution is k6, then the sum of k and all possible values of n is :

  1. 22
  2. 20
  3. 24
  4. 21
Ques. 3JEE Main 2026, 22 January (Evening Shift)

If X = bmatrix x \ y \ z bmatrix is a solution of the system of equations AX = B, where adj A = bmatrix 4 & 2 & 2 \ -5 & 0 & 5 \ 1 & -2 & 3 bmatrix and B = bmatrix 4 \ 0 \ 2 bmatrix, then |x + y + z| is equal to :

  1. 3
  2. 2
  3. 32
  4. 1
Ques. 4JEE Main 2026, 21 January (Evening Shift)

If the system of equations 3x + y + 4z = 3 2x + y - z = -3 x + 2y + z = 4 has no solution, then the value of is equal to:

  1. 13
  2. 4
  3. 19
  4. 23
Ques. 5JEE Main 2025, 8 April (Evening Shift)

Let A = bmatrix 2 & 2+p & 2+p+q \ 4 & 6+2p & 8+3p+2q \ 6 & 12+3p & 20+6p+3q bmatrix. If (adj(adj(3A))) = 2m 3n, m, n N, then m + n is equal to

  1. 22
  2. 20
  3. 24
  4. 26
Ques. 6JEE Main 2025, 7 April (Morning Shift)

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
Ques. 7JEE Main 2025, 7 April (Morning Shift)

Let the system of equations : 2x + 3y + 5z = 9 7x + 3y - 2z = 8 12x + 3y - (4 + )z = 16 - have infinitely many solutions. Then the radius of the circle centred at ( , ) and touching the line 4x = 3y is :

  1. 75
  2. 215
  3. 7
  4. 175
Ques. 8JEE Main 2025, 7 April (Morning Shift)

The number of singular matrices of order 2, whose elements are from the set \2, 3, 6, 9\, is .

Ques. 9JEE Main 2025, 7 April (Evening Shift)

Let the system of equations x + 5y - z = 1 4x + 3y - 3z = 7 24x + y + z = , R, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy 7 x + y + z 77, is :

  1. 4
  2. 5
  3. 3
  4. 6
Ques. 10JEE Main 2025, 3 April (Morning Shift)

If y(x) = vmatrix x & x & x+ x+1 \ 27 & 28 & 27 \ 1 & 1 & 1 vmatrix, x R, then d2ydx2 + y is equal to

  1. 28
  2. 27
  3. -1
  4. 1
Ques. 11JEE Main 2025, 3 April (Morning Shift)

Let A be a matrix of order 3 3 and |A| = 5. If |2 adj(3A adj(2A))| = 2 3 5 , , , N, then + + is equal to

  1. 26
  2. 27
  3. 25
  4. 28

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