Exams › JEE Advanced › Maths

Let A = [a_ij] be a 3x3 matrix where a_ij + a_ji = 0 and each element a_ij belongs to the set {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7}. How many such matrices A are possible?

1. 3370
2. 3300
3. 3375
4. None of these

Correct answer: 3375

Solution

Condition a_ij + a_ji = 0: skew-symmetric matrix. Diagonal: a_ii + a_ii = 0 -> a_ii = 0 (3 elements fixed). Off-diagonal pairs (i<j): a_ij = -a_ji. For each pair, choose a_ij from the valid set. Valid values: a_ij in {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7} and a_ji = -a_ij must also be in this set. Since the set is symmetric about 0, any choice of a_ij from {0, ±1,...,±7} automatically gives a valid a_ji. Set size = 15 (0, ±1, ±2, ±3, ±4, ±5, ±6, ±7 = 1 + 2*7 = 15). Number of off-diagonal pairs in 3x3 = 3 (pairs: (1,2),(1,3),(2,3)). Total matrices = 15³ = 3375.

Related JEE Advanced Maths questions

• Given that A and B are symmetric matrices and they commute (AB = BA), what type of matrix is A^T B?
• Given two matrices A and B satisfying AB = B and BA = A, what is the value of A² + B²?
• Consider the matrix P = [1, 0, 0; 4, 1, 0; 16, 4, 1] and the identity matrix I of size 3. If a matrix Q = [q_(ij)] satisfies P⁵⁰ - Q = I, what is the value of (q₃₁ + q₃₂)/(q₂₁) ?
• Which of the following matrices cannot be expressed as the square of a 3 × 3 matrix with real elements?
• What is the total number of 3 × 3 matrices M, whose elements are chosen from {0, 1, 2}, such that the sum of the diagonal elements of MᵀM equals 5?
• Given the matrix M = [[5/2, 3/2], [−3/2, −1/2]], which of the following represents the value of M raised to the power 2022?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →