 Invertible anticommuting matrices have even dimension and zero traces

For complex invertible matrices $A,B$, $AB=-BA$ implies $B^{-1}AB=-A$ and $A^{-1}BA=-B$. Invariance of matrix trace under matrix similarity gives both zero traces. Invariance of determinant gives $\det A=(-1{)}^n\det A$, and invertibility forces $n$ even. Equivalently, eigenvalues occur in pairs $\lambda ,-\lambda$ with equal algebraic multiplicity, and zero is excluded. Both invertibility assumptions matter: $A=\mathrm{diag}(1,0,0)$ and $B=\mathrm{diag}(0,1,0)$ anticommute in dimension three but are singular and have nonzero traces.