Exams › JEE Advanced › Maths

Consider the matrix P = [ 2 0 0 ] [ 0 2 0 ] [ 0 0 3 ]. Let the transpose of a matrix X be denoted by Xᵀ. Then the number of 3 × 3 invertible matrices Q with integer entries, such that Q⁻¹ = Qᵀ and PQ = QP, is

1. 32
2. 8
3. 16
4. 24

Correct answer: 16

Solution

For Q⁻¹ = Qᵀ, the matrix Q must be orthogonal, meaning QᵀQ = I. Additionally, the condition PQ = QP implies Q must commute with P, which restricts Q to a specific form where it preserves the eigenvalues of P. Considering these constraints and the fact that Q has integer entries, there are exactly 16 such matrices that satisfy both conditions.

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 →