Let $B \in M_{n×n}(\Bbb F)$. Define the functions $L_B$ and $R_B$ by: $L_{B}(A) = BA$ and $R_{B}(A) = AB$. Prove that $\det(L_B) = \big(\det(B)\big)^n$, $\det(R_B) = \big(\det(B)\big)^n$.
$L_B$ and $R_B$ are linear transformations, thus can be represented by matrices. So the determinant of those functions exists.