If A is any square matrix such that `A+I/2` and `A-I/2` are orthogonal matrices, then

Options

A. A is symmetric
B. A is skew-symmetric
C. `A^(2)=(3l)/(4)`
D. `A^(2)=(-3l)/(4)`

Answer

To solve the problem, we need to analyze the conditions given for the matrices \( B = A + \frac{I}{2} \) and \( C = A - \frac{I}{2} \) being orthogonal matrices.

### Step-by-step Solution:

1. **Understanding Orthogonal Matrices**:
- A matrix \( M \) is orthogonal if \( M M^T = I \), where \( M^T \) is the transpose of \( M \) and \( I \) is the identity matrix.

2. **Setting Up the Equations**:
- For \( B = A + \frac{I}{2} \):
\[ B B^T = (A + \frac{I}{2})(A^T + \frac{I}{2}) = I \]
- For \( C = A - \frac{I}{2} \):
\[ C C^T = (A - \frac{I}{2})(A^T - \frac{I}{2}) = I \]

3. **Expanding the First Equation**:
- Expanding \( B B^T \):
\[ (A + \frac{I}{2})(A^T + \frac{I}{2}) = A A^T + A \cdot \frac{I}{2} + \frac{I}{2} A^T + \frac{I}{4} = I \]
- This simplifies to:
\[ A A^T + \frac{1}{2}(A + A^T) + \frac{I}{4} = I \]

4. **Expanding the Second Equation**:
- Expanding \( C C^T \):
\[ (A - \frac{I}{2})(A^T - \frac{I}{2}) = A A^T - A \cdot \frac{I}{2} - \frac{I}{2} A^T + \frac{I}{4} = I \]
- This simplifies to:
\[ A A^T - \frac{1}{2}(A + A^T) + \frac{I}{4} = I \]

5. **Setting the Two Equations Equal**:
- From the two expanded equations, we have:
\[ A A^T + \frac{1}{2}(A + A^T) + \frac{I}{4} = I \quad \text{(1)} \]
\[ A A^T - \frac{1}{2}(A + A^T) + \frac{I}{4} = I \quad \text{(2)} \]

6. **Subtracting the Two Equations**:
- Subtract equation (2) from equation (1):
\[ \left(A A^T + \frac{1}{2}(A + A^T) + \frac{I}{4}\right) - \left(A A^T - \frac{1}{2}(A + A^T) + \frac{I}{4}\right) = 0 \]
- This simplifies to:
\[ A + A^T = 0 \]
- Thus, \( A^T = -A \), meaning \( A \) is skew-symmetric.

7. **Finding \( A^2 \)**:
- From the first equation (1):
\[ A A^T + \frac{1}{2}(A + A^T) + \frac{I}{4} = I \]
- Since \( A + A^T = 0 \):
\[ A A^T + \frac{I}{4} = I \]
- Rearranging gives:
\[ A A^T = I - \frac{I}{4} = \frac{3I}{4} \]

8. **Using the Property of Skew-Symmetric Matrices**:
- For skew-symmetric matrices, \( A A^T = -A^2 \):
\[ -A^2 = \frac{3I}{4} \]
- Therefore:
\[ A^2 = -\frac{3I}{4} \]

### Final Result:
Thus, we conclude that \( A \) is a skew-symmetric matrix and \( A^2 = -\frac{3I}{4} \).