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

To resolve this, we must determine which matrix type can be simultaneously symmetric and skew-symmetric.

1. Defining Symmetric and Skew-Symmetric Matrices:
A matrix \( A \) is symmetric when:

\( A^T = A \)

A matrix \( A \) is skew-symmetric when:

\( A^T = -A \)
For a matrix to be both symmetric and skew-symmetric, the following must hold:

\( A = A^T = -A \Rightarrow A = -A \)
This leads to:

\( 2A = 0 \Rightarrow A = 0 \)
Therefore, the only matrix fulfilling both conditions is the null matrix, composed entirely of zeros.

2. Examining Each Option:
(A) Unit Matrix → Impossible. It is symmetric but not skew-symmetric.
(B) Diagonal Matrix → May be symmetric, but not necessarily skew-symmetric.
(C) Null Matrix → Satisfies both \( A = A^T \) and \( A = -A \). → Correct
(D) Row Matrix → Typically not square, thus ineligible for symmetric/skew-symmetric classification.

3. Conclusion:
The null matrix is the sole matrix possessing both symmetric and skew-symmetric properties.

Final Answer:
The correct classification is Null Matrix.