linear operator, determinant

Hi, I am having trouble with the following exercise.

Let V be the vector space of all nxn matrices over the field of complex numbers, and let B be a fixed nxn matrix over C. Define a linear operator M_B on V by M_B(A) = BAB*. Show that det M_B = |det B|^{2n}.

The determinant of M_B is equal to that of the matrix P representing M_B in an ordered base S for V. Let E_{xy} be a matrix in V such that [E_{xy}]_{ij} = 1 if (i,j)=(x,y), 0 otherwise. Let S = {E_{11}, E_{12}, ...., E_{1n}, E_{21}, ...., E_{2n}, .........., E_{n1}, ..., E_{nn}} be an ordered base of V.
First of all I start to determine the generic element ij of BE_{xy}B*:
[BE_{xy}B*]_{ij}
= Sum_k [B E_{xy}]_{ik}{B*}_{kj} =
= Sum_k ~B_{jk}[B E_{xy}]_{ik} =
= Sum_k ~B_{jk}Sum_h B_{ih}[E_{xy}]_{hk} =
= ~B_{jy}Sum_h B_{ih}[E_{xy}]_{hy} =
= ~B_{jy}B_{ix}[E_{xy}]_{xy} =
= ~B_{jy}B_{ix}
Now I am in big trouble. How can I write P?
I could write
P_{st} = ~B_{jy}B_{ix}
where
s = (i-1)n + j and
t = (x-1)n + y
but what should I do now?
Should I write P explicitly hoping for a miracle or should I use the general determinant formula (the sum extended over the permutation) expecting some simplification?

Kiuhnm
.