If A + B = [ array cr 1 & 2 - 2 & 1 array ] where A is symmetric and B is skew-symmetric matrix, then the matrix (A⁻¹ B+A B⁻¹ ) at = 6 is given by

Step-by-step solution

Given, A+B= [ array cc 1 & 2 - 2 & 1 array ] ...(i) A^T+B^T= [ array cc 1 & - 2 2 & 1 array ] Since A is symmetric and B is skew symmetric aligned & A^T=A, B^T=-B & A-B= [ array cc 1 & - 2 2 & 1 array ]...(ii) aligned Adding (i) and (ii), we get aligned 2 A & = [ array cc 1 & 2 - 2 & 1 array ]+ [ array cc 1 & - 2 2 & 1 array ] 2 A & = [ array ll 2 & 0 0 & 2 array ] A & = [ array ll 1 & 0 0 & 1 array ] aligned Now, subtracting (ii) from (i), we get aligned & 2 B= [ array cc 1 & 2 - 2 & 1 array ]- [ array cc 1 & - 2