Exams › JEE Advanced › Maths

Let A be a symmetric matrix and B be a skew-symmetric matrix such that AB = BA. Which of the following is/are correct?

1. (A - B)^(-1) * (A + B) is an orthogonal matrix when (A - B) is non-singular
2. (A + B)^(-1) * (A - B) is an orthogonal matrix when (A + B) is non-singular
3. det((A - B)^(-1) * (A + B)) = 1 and det((A + B)^(-1) * (A - B)) = -1
4. det((A - B)^(-1) * (A + B)) = -1 and det((A + B)^(-1) * (A - B)) = 1

Correct answer: (A - B)^(-1) * (A + B) is an orthogonal matrix when (A - B) is non-singular

Solution

A^T = A (symmetric), B^T = -B (skew-symmetric). Let M = (A-B)^(-1)(A+B). M^T = (A+B)^T * ((A-B)^T)^(-1) = (A+B)^T * (A-B)^(-T) = (A-B)(A+B)^(-1). For M^T*M = I: (A-B)(A+B)^(-1)*(A-B)^(-1)(A+B) = (A-B) * [(A+B)^(-1)*(A-B)^(-1)] * (A+B). Using AB=BA, we get (A+B) and (A-B) commute, so [(A+B)(A-B)]^(-1) = [(A-B)(A+B)]^(-1). Thus M^T*M = (A-B)(A+B)^(-1)(A-B)^(-1)(A+B) = (A-B)*(A+B)^(-1)*(A-B)^(-1)*(A+B). Since AB=BA: (A+B)(A-B) = A²-AB+BA-B² = A²-B² = (A-B)(A+B). So they commute! Therefore (A+B)^(-1)*(A-B)^(-1) = [(A-B)(A+B)]^(-1) = [(A+B)(A-B)]^(-1) = (A-B)^(-1)*(A+B)^(-1). M^T*M = (A-B)*(A-B)^(-1)*(A+B)^(-1)*(A+B) = I*I = I. So M is orthogonal. Similarly (A+B)^(-1)(A-B) is orthogonal.

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 →