IMC 2011 · Problem 2

Does there exist a real $3 \times 3$ matrix $A$ such that $\operatorname{tr}(A) = 0$ and $A^2 + A^t = I$? ($\operatorname{tr}(A)$ denotes the trace of $A$, $A^t$ is the transpose of $A$, and $I$ is the identity matrix.)