If A is a symmetric matrix, B is a skew-symmetric matrix, `A+B` is nonsingular and `C=(A+B)^(-1) (A-B)`, then prove that
(i) `C^(T) (A+B) C=A+B` (ii) `C^(T) (A-B)C=A-B`
(iii) `C^(T)AC=A`

Explanation

To solve the problem step by step, we will prove the three statements given in the question using the properties of symmetric and skew-symmetric matrices.

### Given:
- \( A \) is a symmetric matrix: \( A^T = A \)
- \( B \) is a skew-symmetric matrix: \( B^T = -B \)
- \( A + B \) is nonsingular (invertible)
- \( C = (A + B)^{-1} (A - B) \)

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