Question

Which of the following can be both a symmetric and skew-symmetric matrix?

• A. Unit Matrix
• B. Diagonal Matrix
• C. Null Matrix
• D. Row Matrix

Hint:

The null matrix is the only matrix that can be both symmetric and skew-symmetric since \( 0 = 0^T \) and \( 0 = -0^T \).

Answer 1

The Correct Option is C

Step 1: Understanding symmetric and skew-symmetric matrices.
A matrix is symmetric if \( A = A^T \), meaning the matrix is equal to its transpose. A matrix is skew-symmetric if \( A = -A^T \), meaning the matrix is equal to the negative of its transpose. Step 2: Finding the matrix that satisfies both conditions.
The only matrix that satisfies both symmetric and skew-symmetric properties is the null matrix, because: \[ 0 = 0^T \quad \text{(symmetric)} \quad \text{and} \quad 0 = -0^T \quad \text{(skew-symmetric)}. \] Thus, the null matrix is both symmetric and skew-symmetric.