If A is any square matrix, then

Options

A. `A+A^(T)` is skew symmetric
B. `A-A^(T)` is symmetric
C. `A A^(T)` is symmetric
D. `A A^(T)` is skew symmetric

Answer

To solve the problem step by step, we need to analyze the properties of the square matrix \( A \) and its transpose \( A^T \). We will check the validity of the given options one by one.

### Step 1: Check Option 1 - Is \( A + A^T \) skew symmetric?

1. **Definition**: A matrix \( B \) is skew symmetric if \( B^T = -B \).
2. **Calculate \( (A + A^T)^T \)**:
\[ (A + A^T)^T = A^T + (A^T)^T = A^T + A \]
This simplifies to:
\[ A + A^T \]
3. **Conclusion**: Since \( (A + A^T)^T = A + A^T \), it shows that \( A + A^T \) is symmetric, not skew symmetric.

**Result for Option 1**: **Incorrect**.

### Step 2: Check Option 2 - Is \( A - A^T \) symmetric?

1. **Definition**: A matrix \( B \) is symmetric if \( B^T = B \).
2. **Calculate \( (A - A^T)^T \)**:
\[ (A - A^T)^T = A^T - (A^T)^T = A^T - A \]
This simplifies to:
\[ - (A - A^T) \]
3. **Conclusion**: Since \( (A - A^T)^T = - (A - A^T) \), it shows that \( A - A^T \) is skew symmetric, not symmetric.

**Result for Option 2**: **Incorrect**.

### Step 3: Check Option 3 - Is \( A \cdot A^T \) symmetric?

1. **Definition**: A matrix \( B \) is symmetric if \( B^T = B \).
2. **Calculate \( (A \cdot A^T)^T \)**:
\[ (A \cdot A^T)^T = (A^T)^T \cdot A^T = A \cdot A^T \]
3. **Conclusion**: Since \( (A \cdot A^T)^T = A \cdot A^T \), it shows that \( A \cdot A^T \) is symmetric.

**Result for Option 3**: **Correct**.

### Final Conclusion:
- Option 1: Incorrect
- Option 2: Incorrect
- Option 3: Correct