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 solve the problem, we need to identify which type of matrix can be both symmetric and skew-symmetric.

1. Understanding Symmetric and Skew-Symmetric Matrices: 
A matrix \( A \) is symmetric if:

\( A^T = A \)

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

\( A^T = -A \)
Now, if a matrix is both symmetric and skew-symmetric, then:

\( A = A^T = -A \Rightarrow A = -A \)
This implies:

\( 2A = 0 \Rightarrow A = 0 \)
So, the only matrix that satisfies both conditions is the null matrix (all elements are zero).

2. Evaluating Each Option:
(A) Unit Matrix → Not possible. It's symmetric but not skew-symmetric.
(B) Diagonal Matrix → Could be symmetric, but not necessarily skew-symmetric.
(C) Null Matrix → Satisfies both \( A = A^T \) and \( A = -A \). → Correct
(D) Row Matrix → Not necessarily square, and thus not even eligible for symmetric/skew-symmetric.

3. Conclusion:
The null matrix is the only matrix that is both symmetric and skew-symmetric.

Final Answer:
The correct answer is Null Matrix.