📘 Question
Write a \(2 \times 2\) matrix which is both symmetric and skew-symmetric.
✏️ Step-by-Step Solution
Step 1: Recall definitions
- Symmetric matrix: \(A^T = A\)
- Skew-symmetric matrix: \(A^T = -A\)
Step 2: Combine both conditions
If a matrix is both symmetric and skew-symmetric, then:
\[
A^T = A \quad \text{and} \quad A^T = -A
\]
So,
\[
A = -A
\]
Step 3: Solve the condition
\[
2A = 0 \Rightarrow A = 0
\]
Step 4: Write the matrix
\[
A = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}
\]
✅ Final Answer
\[
\boxed{\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}}
\]
💡 Key Concept
The only matrix that is both symmetric and skew-symmetric is the zero matrix, because it satisfies:
- \(A^T = A\)
- \(A^T = -A\)
This is possible only when all elements are zero.