Question:

Step 1: State the given properties of matrices A and B, and the definition of C. We are given that A is a symmetric matrix, which means its transpose is equal to itself: $$A^T = A$$ We are also given that B is a skew-symmetric matrix, which means its transpose is the negative of itself: $$B^T = -B$$ The matrix C is defined as: $$C = (A + B)^{-1}(A - B)$$ It is specified that $A+B$ is non-singular, which ensures that its inverse $(A+B)^{-1}$ exists. Step 2: Calculate the transpose of C, $C^T$. We use the property of matrix transposes, $(XY)^T = Y^T X^T$, and the property $(X^{-1})^T = (X^T)^{-1}$: $$C^T = \left((A + B)^{-1}(A - B)\right)^T$$ Applying the transpose product rule: $$C^T = (A - B)^T \left((A + B)^{-1}\right)^T$$ Further applying transpose rules $(X-Y)^T = X^T - Y^T$ and $(X+Y)^T = X^T + Y^T$: $$C^T = (A^T - B^T) ((A + B)^T)^{-1}$$ Step 3: Substitute the properties of symmetric and skew-symmetric matrices into the expression for $C^T$. Now, substitute $A^T = A$ and $B^T = -B$ into the expression for $C^T$: $$C^T = (A - (-B)) ((A^T + B^T))^{-1}$$ $$C^T = (A + B) (A + (-B))^{-1}$$ $$C^T = (A + B) (A - B)^{-1}$$ Step 4: Substitute the expressions for $C^T$ and $C$ into the required expression $C^T(A - B)C$. We need to evaluate $C^T(A - B)C$. Substitute the expression for $C^T$ derived in Step 3 and the given expression for $C$: $$C^T(A - B)C = \left((A + B)(A - B)^{-1}\right) (A - B) \left((A + B)^{-1}(A - B)\right)$$ Step 5: Simplify the expression using matrix inverse properties. We use the fundamental property that a matrix multiplied by its inverse yields the identity matrix, i.e., $X X^{-1} = I$, and $IX = X$: $$C^T(A - B)C = (A + B) \left((A - B)^{-1}(A - B)\right) (A + B)^{-1}(A - B)$$ The term $(A - B)^{-1}(A - B)$ simplifies to the identity matrix $I$: $$C^T(A - B)C = (A + B) I (A + B)^{-1}(A - B)$$ Since multiplying by the identity matrix does not change a matrix ($XI = X$): $$C^T(A - B)C = (A + B) (A + B)^{-1}(A - B)$$ Now, the term $(A + B)(A + B)^{-1}$ also simplifies to the identity matrix $I$: $$C^T(A - B)C = I (A - B)$$ Finally, multiplying by the identity matrix gives: $$C^T(A - B)C = A - B$$ Step 6: State the final answer. The simplified expression for $C^T(A - B)C$ is $A - B$. Comparing this result with the given options, it matches Option 2. The final answer is $\boxed{\text{A - B}}$.