If A and B are symmetric matrices of same order, then
STATEMENT-1: A+B is skew - symmetric matrix.
STATEMENT -2 : AB-BA is skew - symmetric matrix.
STATEMENT-3 A-B is skew - symmetric matrix .

Options

A. T T T
B. F T F
C. F T T
D. F F F

Answer

To solve the problem, we need to analyze each statement regarding symmetric matrices A and B.

### Step-by-Step Solution:

**Given:**
- A and B are symmetric matrices of the same order.
- A symmetric matrix satisfies the condition \( A = A^T \) and \( B = B^T \).

**Statement 1: \( A + B \) is a skew-symmetric matrix.**

1. **Check if \( A + B \) is skew-symmetric:**
- We need to find the transpose of \( A + B \):
\[ (A + B)^T = A^T + B^T \]
- Since A and B are symmetric, we have:
\[ A^T = A \quad \text{and} \quad B^T = B \]
- Therefore:
\[ (A + B)^T = A + B \]
- This shows that \( A + B \) is symmetric, not skew-symmetric.

**Conclusion for Statement 1:** **False**

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**Statement 2: \( AB - BA \) is a skew-symmetric matrix.**

1. **Check if \( AB - BA \) is skew-symmetric:**
- We need to find the transpose of \( AB - BA \):
\[ (AB - BA)^T = (AB)^T - (BA)^T \]
- Using the property of transpose:
\[ (AB)^T = B^T A^T = BA \quad \text{and} \quad (BA)^T = A^T B^T = AB \]
- Therefore:
\[ (AB - BA)^T = BA - AB = -(AB - BA) \]
- This shows that \( AB - BA \) is skew-symmetric.

**Conclusion for Statement 2:** **True**

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**Statement 3: \( A - B \) is a skew-symmetric matrix.**

1. **Check if \( A - B \) is skew-symmetric:**
- We need to find the transpose of \( A - B \):
\[ (A - B)^T = A^T - B^T \]
- Since A and B are symmetric:
\[ (A - B)^T = A - B \]
- This shows that \( A - B \) is symmetric, not skew-symmetric.

**Conclusion for Statement 3:** **False**

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### Final Summary:
- **Statement 1:** False
- **Statement 2:** True
- **Statement 3:** False