Expressing Determinant as Permanent One major problem in TCS is the problem of expressing a permanent as a determinant. I was reading Agrawal's paper Determinant Versus Permanent and in one paragraph he claims the reverse problem is easy.


  It is easy to see that the determinant of a matrix $X$ can be expressed as the permanent of a related matrix $Xˆ$ whose entries are 0, 1, or $x_{i,j}$ s and which is of size $O(n)$ (set up entries of Xˆ such that det $Xˆ$ = det $X$ and the product corresponding to every permutation that has an even cycle is zero).


First of all, I don't think 0, 1, and $x_{i,j}$ variables are enough because we would be missing negative terms. But even if we allowed -1 and $-x_{i,j}$ variables as well, I don't see why the growth in size can be made linear. Could someone please explain the construction to me?
I think this may have been a typo in Agrawal's paper. The best I know is how to write an $n \times n$ determinant as a projection of an $O(n^3)$-sized permanent, by writing the determinant as an algebraic branching program (and I think this is currently the best known). See the comments on this answer.