IMC 2011 Problem 2

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

Proposed by Moubinool Omarjee, Paris