Consider a linear operator $A$ acting on a vector space $V$ of finite dimension $N$ and a linear operator $B$ acting on a vector space $W$ of finite dimension $M$. Assuming that orthonormal bases exist in $V$ and in $W$, and that the operators are represented in terms of the matrices of their coefficients in these bases, prove that the determinant of the tensor product $A \otimes B$ is given by $\operatorname{det}(A \otimes B)=$ $(\operatorname{det} A)^{M} \cdot(\operatorname{det} B)^{N}$.