A classical matrix equation

Suppose there exists two matrices $A,B \in M_{n}(\R)$ such that $AB - BA = I_{n}$. By linearity and the cyclic property of the Trace, we get
$$n=\mathrm{Tr}(I_n)=\text{tr}(AB - BA) = \text{tr}(AB) - \text{tr}(BA) = 0$$an absurdity.