Trace and determinant of composition of a left-multiplication and a right-multiplication on a space of matrices

Determine the trace and determinant of the linear operator (on the space $\mathbb{F^{n\times n}}$) that sends the matrix $M\to AMB$ where $A$ and $B$ are $n\times n$ matricies

Let $L_A$ be left multiplication by $A$. Then you can split $F^{n\times n}$ into the direct sum of $L_A$-invariant $n$ copies of $F^n$ (the columns of $M$), on each of which $L_A$ acts by the usual multiplication of a matrix by a column vector. Hence $\det(L_A)=[\det(A)]^n$. Similarly, $\det(R_B)=[\det(B)]^n$. Hence $\det(L_AR_B)=\det(L_A)\det(R_B)=[\det(A)]^n[\det(B)]^n.$ To compute $tr(L_AL_B)$, refine the previous argument to $L_AR_B=A\otimes B^*$ (thinking of $F^{n\times n}=Hom(F^n, F^n)=F^n\otimes (F^n)^*$), hence $tr(L_AR_B)=tr(A)tr(B)$.