Transpose_of_Matrix_Product_v2

Let $\mathbf A$ and $\mathbf B$ be matrices over a commutative ring such that $\mathbf A \mathbf B$ is defined. Then $\mathbf B^\intercal \mathbf A^\intercal$ is defined, and $\paren {\mathbf A \mathbf B}^\intercal = \mathbf B^\intercal \mathbf A^\intercal$ where $\mathbf X^\intercal$ is the transpose of $\mathbf X$.