Everyone knows that for any two square matrices $A$ and $B$ (with coefficients in a commutative ring) that $$tr(AB) \operatorname{tr}(AB) = tr(BA).$$\operatorname{tr}(BA).$$I once thought that this implied (via induction) that the trace of a product of any finite number of matrices was independent of the order they are multiplied. 1 [made Community Wiki] Everyone knows that for any two square matrices A and B (with coefficients in a commutative ring) that$$tr(AB) = tr(BA).