Matrices & Determinants
Symmetric and Skew-symmetric matrices
Grade Class 12
Question:
Let A and B be 3 x 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A^2 B^2 - B^2 A^2)X = 0, where X is a 3 x 1 column matrix of unknown variables and 0 is a 3 x 1 null matrix, has :
(1) no solution
(2) exactly two solutions
(3) infinitely many solutions
(4) a unique solution
Step-by-Step Solution
Key Concept: The matrix M = A^2 B^2 - B^2 A^2 is skew-symmetric. For any skew-symmetric matrix M of odd order (3x3), the determinant is 0. Thus, the system MX = 0 has infinitely many solutions.
Let M = A^2 B^2 - B^2 A^2. Since A is symmetric (A^T = A) and B is skew-symmetric (B^T = -B), we have M^T = (A^2 B^2 - B^2 A^2)^T = (B^2)^T (A^2)^T - (A^2)^T (B^2)^T = B^2 A^2 - A^2 B^2 = -(A^2 B^2 - B^2 A^2) = -M. Thus, M is a skew-symmetric matrix of order 3. The determinant of a skew-symmetric matrix of odd order is 0. Therefore, det(M) = 0. The system MX = 0 has a non-trivial solution, implying infinitely many solutions.
Correct Answer: 3