📘 Question
If a matrix \(A\) is both symmetric and skew-symmetric, then:
(a) \(A\) is a diagonal matrix
(b) \(A\) is a zero matrix
(c) \(A\) is a scalar matrix
(d) \(A\) is a square matrix
✏️ Step-by-Step Solution
Step 1: Use definitions
- Symmetric matrix: \(A^T = A\)
- Skew-symmetric matrix: \(A^T = -A\)
Step 2: Combine both
\[
A^T = A \quad \text{and} \quad A^T = -A
\]
So,
\[
A = -A
\]
Step 3: Solve
\[
2A = 0 \Rightarrow A = 0
\]
Step 4: Conclusion
Thus, matrix \(A\) must be the zero matrix.
✅ Final Answer
\[
\boxed{(b)\; \text{zero matrix}}
\]
💡 Key Concept
Only the zero matrix satisfies both:
- \(A^T = A\)
- \(A^T = -A\)