`A,B,C` are three matrices of the same order such that any two are symmetric and the `3^(rd)` one is skew symmetric. If `X=ABC+CBA` and `Y=ABC-CBA`, then `(XY)^(T)` is

Options

A. symmetric
B. skew symmetric
C. `I-XY`
D. `-YX`

Explanation

To solve the problem, we need to find the transpose of the product of two matrices \(X\) and \(Y\) defined as:

- \(X = ABC + CBA\)
- \(Y = ABC - CBA\)

Given that matrices \(A\) and \(B\) are symmetric and matrix \(C\) is skew-symmetric, we can use the properties of transposes to find \((XY)^T\).

### Step-by-Step Solution:

1. **Identify the properties of the matrices:**
- Since \(A\) and \(B\) are symmetric, we have:
\[ A^T = A \quad \text{and} \quad B^T = B \]
- Since \(C\) is skew-symmetric, we have:
\[ C^T = -C \]

2. **Find \(Y^T\):**
\[ Y = ABC - CBA \]
Taking the transpose:
\[ Y^T = (ABC - CBA)^T = (CBA)^T - (ABC)^T \]
Using the property \((AB)^T = B^T A^T\):
\[ Y^T = A^T B^T C^T - C^T B^T A^T \]
Substituting the properties of \(A\), \(B\), and \(C\):
\[ Y^T = A B (-C) - (-C) B A = -ABC + CBA = Y \]
Thus, we conclude:
\[ Y^T = Y \]

3. **Find \(X^T\):**
\[ X = ABC + CBA \]
Taking the transpose:
\[ X^T = (ABC + CBA)^T = (CBA)^T + (ABC)^T \]
Again using the transpose property:
\[ X^T = A^T B^T C^T + C^T B^T A^T \]
Substituting the properties of \(A\), \(B\), and \(C\):
\[ X^T = A B (-C) + (-C) B A = -ABC - CBA = -(ABC + CBA) = -X \]
Thus, we conclude:
\[ X^T = -X \]

4. **Calculate \((XY)^T\):**
Using the property \((XY)^T = Y^T X^T\):
\[ (XY)^T = Y^T X^T = Y (-X) = -YX \]

5. **Final Result:**
Therefore, we conclude that:
\[ (XY)^T = -YX \]