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It seems "common knowledge" that the following holds:

Let T be a linear transformation on nxn matrices with complex coefficients that preserves the determinant. Then there exists matrices U and V whose product has determinant 1 such that one of the following holds:

a) For any matrix A we have T(A)=UAV
b) For any matrix A we have T(A)=UBV where B is the transpose of A

It seems quite reasonable, but as far as "common knowledge" goes, I have no clue right now on how to prove such a thing?

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# Linear transformation that preserves the determinant.

It seems "common knowledge" that the following holds:

Let T be a linear transformation on nxn matrices with complex coefficients that preserves the determinant. Then there exists matrices U and V whose product has determinant 1 such that one of the following holds:

a) For any matrix A we have T(A)=UAV b) For any matrix A we have T(A)=UBV where B is the transpose of A

It seems quite reasonable, but as far as "common knowledge" goes, I have no clue right now on how to prove such a thing?