If A is a square matrix such that `A A^T=I=A^TA`, then A is

Options

A. a symmetric matrix
B. a skew-symmetric matrix
C. a diagonal matrix
D. an orthogonal matrix.

Answer

To determine the nature of the square matrix \( A \) given that \( A A^T = I \) and \( A^T A = I \), we can follow these steps:

### Step 1: Understand the Definitions
We need to recall the definitions of the types of matrices mentioned:
- **Symmetric Matrix**: A matrix \( A \) is symmetric if \( A^T = A \).
- **Skew-Symmetric Matrix**: A matrix \( A \) is skew-symmetric if \( A^T = -A \).
- **Diagonal Matrix**: A matrix is diagonal if all its non-diagonal elements are zero.
- **Orthogonal Matrix**: A matrix \( A \) is orthogonal if \( A A^T = I \) and \( A^T A = I \).

### Step 2: Analyze the Given Conditions
We are given that:
\[ A A^T = I \]
\[ A^T A = I \]

These conditions indicate that \( A \) satisfies the definition of an orthogonal matrix.

### Step 3: Verify the Options
Now, let's check the options provided:
1. **Symmetric Matrix**: Not necessarily true since \( A^T \) does not have to equal \( A \).
2. **Skew-Symmetric Matrix**: Not true since \( A^T \) does not equal \( -A \).
3. **Diagonal Matrix**: Not necessarily true; a diagonal matrix is a specific case and does not encompass all orthogonal matrices.
4. **Orthogonal Matrix**: This is true based on our analysis.

### Conclusion
Since \( A A^T = I \) and \( A^T A = I \), we conclude that \( A \) is an orthogonal matrix.

### Final Answer
Thus, the answer is that \( A \) is an **orthogonal matrix**.

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