Show that \((A \otimes B)\left(A^{\prime} \otimes B^{\prime}ight)=\left(A A^{\prime}ight) \otimes\left(B B^{\prime}ight)\). For simplicity let the matrices \(A, A^{\prime}, B, B^{\prime}\) all be \(n \times n\). The general requirement on matrix dimensions for this relationship is that the number of columns in \(A\) must equal the number of rows in \(A^{\prime}\) to allow matrix multiplication \(A A^{\prime}\), and similarly for \(B B^{\prime}\).