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>(A - B)C =
Step-by-Step Solution
Key Concept: Use the properties of transpose and the given definitions of symmetric (A^T = A) and skew-symmetric (B^T = -B) matrices to simplify the expression C^T(A - B)C.
Given A<sup>T</sup> = A and B<sup>T</sup> = -B. C = (A + B)<sup>-1</sup>(A - B). Then C<sup>T</sup> = (A - B)<sup>T</sup>((A + B)<sup>-1</sup>)<sup>T</sup> = (A<sup>T</sup> - B<sup>T</sup>)((A + B)<sup>T</sup>)<sup>-1</sup> = (A + B)(A<sup>T</sup> + B<sup>T</sup>)<sup>-1</sup> = (A + B)(A - B)<sup>-1</sup>. Now, C<sup>T</sup>(A - B)C = (A + B)(A - B)<sup>-1</sup>(A - B)(A + B)<sup>-1</sup>(A - B) = (A + B)I(A + B)<sup>-1</sup>(A - B) = (A + B)(A + B)<sup>-1</sup>(A - B) = I(A - B) = A - B.
Correct Answer: 2