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The cofactor formula for computing the determinant of a matrix is applicable when elements of the matrix are from a commutative ring. However, the complexity of this method is extremely high and I believe there should be a better solution.

If it is a matrix over a PID, I believe the method of elimination like Smith normal form may be applicable. But if have a more general ring which is not a PID, are there known methods to do something similar?

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    @joriki That's e$x$actl$y$ what I'm trying to do. I'm just hoping there could be a systematic way to do so :)2012-08-16

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