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Let A be a square matrix. Then which of ...

Let A be a square matrix. Then which of the following is not a symmetric matrix -

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To determine which of the given matrices is not a symmetric matrix, we need to analyze the properties of symmetric and skew-symmetric matrices. ### Step-by-Step Solution: 1. **Understand the Definition of Symmetric Matrix**: A matrix \( A \) is symmetric if \( A^T = A \), where \( A^T \) is the transpose of \( A \). 2. **Understand the Definition of Skew-Symmetric Matrix**: A matrix \( A \) is skew-symmetric if \( A^T = -A \). 3. **Analyze Each Option**: Let's denote the options as follows: - Option 1: \( A + A^T \) - Option 2: \( A - A^T \) - Option 3: \( A^T A \) 4. **Check Option 1: \( A + A^T \)**: - Compute the transpose: \[ (A + A^T)^T = A^T + (A^T)^T = A^T + A = A + A^T \] - Since \( (A + A^T)^T = A + A^T \), this matrix is symmetric. 5. **Check Option 2: \( A - A^T \)**: - Compute the transpose: \[ (A - A^T)^T = A^T - (A^T)^T = A^T - A \] - This can be rewritten as: \[ (A - A^T)^T = - (A - A^T) \] - Since \( (A - A^T)^T = - (A - A^T) \), this matrix is skew-symmetric, hence not symmetric. 6. **Check Option 3: \( A^T A \)**: - Compute the transpose: \[ (A^T A)^T = A^T (A^T)^T = A^T A \] - Since \( (A^T A)^T = A^T A \), this matrix is symmetric. ### Conclusion: The matrix that is not symmetric is **Option 2: \( A - A^T \)**.
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