If A is a symmetric and B skew symmetric matrix and A + B is non singular and C = (A + B)<sup>-1</sup>(A - B) then C<sup>T</sup> AC =
Step-by-Step Solution
Key Concept: Use the properties of transpose (A^T = A, B^T = -B) and the definition of C to simplify C^T AC.
Given A^T = A and B^T = -B. C = (A+B)^-1(A-B). C^T = (A-B)^T ((A+B)^-1)^T = (A^T - B^T)(A^T + B^T)^-1 = (A+B)(A-B)^-1. Then C^T AC = (A+B)(A-B)^-1 A (A+B)^-1 (A-B). Since A and B commute in this specific structure, it simplifies to A - B.
Correct Answer: 2