If A and B are symmetric matrices of same order such that AB + BA = X and AB - BA = Y , then ( XY )^ T =

Step-by-step solution

aligned & Since X Y=(A B+B A)(A B-B A) & =( AB ) AB +( BA )( AB )-( AB )( BA )-( BA )( BA ) & Now (x y)^ T =((A B) (A B))^ T +(B A A B)^ T -(A B B A)^ T & =(A B)^ T ( AB )^ T +( AB )^ T ( BA )-( BD )^ T ( AB ) T & -( BA )^ T ( BA )^ T & = (B^T A^T ) (B^T A^T )+ (B^T A^T ) (A^T B^T ) & - (A^T B^T ) (B^T A^T )- (A^T B^T ) (A^T B^T ) & =( BA )( BA )+( BA )( BA )-( AB )( BA )-( AB )( AB ) & = BA ( BA + AB )- AB ( BA + AB ) & =( BA - AB )( BA + AB ) & =- yx & aligned