Matrix that is Both Symmetric and Skew-Symmetric

Condition:

\[ A^T = A \quad \text{(symmetric)} \]

\[ A^T = -A \quad \text{(skew-symmetric)} \]

Step: Combine Both

\[ A = -A \Rightarrow 2A = 0 \Rightarrow A = 0 \]

Final Answer:

\[ A = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

Conclusion:

The zero matrix is the only matrix which is both symmetric and skew-symmetric.