Let A and B be 3 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 3 1 column matrix of unknown variables and O is a 3 1 null matrix, has

Step-by-step solution

Let P=A^2 B^2-B^2 A^2 aligned P ^ T & = ( A ^2 ~B ^2- B ^2 ~A ^2 )^ T & = ( A ^2 ~B ^2 )^ T - ( 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 ^ T = A and B ^ T =- B ] & =- ( A ^2 ~B ^2- B ^2 ~A ^2 ) & =- P aligned P is a skew - symmetric matrix. det ( P )=0 The given system of equations has infinitely many solutions.