Let A be `3xx3` matrix such that A is orthogonal idempotent then

Options

A. A must be symmetric
B. det(A) =-I
C. A +`A^(-1) =1`
D. none of these

Answer

To solve the problem, we need to analyze the properties of the matrix \( A \) given that it is both orthogonal and idempotent.

### Step-by-Step Solution:

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

2. **Understanding Idempotent Matrix**:
A matrix \( A \) is idempotent if \( A^2 = A \).

3. **Combining Properties**:
Since \( A \) is both orthogonal and idempotent, we can write:
\[ A^T A = I \quad \text{(orthogonal)} \]
\[ A^2 = A \quad \text{(idempotent)} \]

4. **Using Idempotent Property**:
From the idempotent property \( A^2 = A \), we can substitute \( A \) into the orthogonal property:
\[ A^T A = I \implies A^T A = A \quad \text{(since \( A^2 = A \))} \]

5. **Taking Transpose**:
Taking the transpose of both sides of \( A^T A = I \):
\[ (A^T A)^T = I^T \]
Using the property of transpose, we have:
\[ A^T A^T = I \implies A A^T = I \]

6. **Conclusion from Orthogonality**:
Since both \( A^T A = I \) and \( A A^T = I \) hold true, this means \( A \) is indeed an orthogonal matrix.

7. **Identifying the Matrix**:
Now, since \( A \) is both orthogonal and idempotent, we can conclude that:
\[ A = I \]
because the only orthogonal idempotent matrix in \( \mathbb{R}^{3 \times 3} \) is the identity matrix.

8. **Symmetry of Matrix**:
Since \( A = I \), we can also state that \( A \) is symmetric because the identity matrix is symmetric:
\[ A^T = A \]

### Final Answer:
Thus, the matrix \( A \) is the identity matrix \( I \), and it is symmetric.