In the paper The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?, the authors report a bug in Mathematica (which is still present in the version 10) that happens when computing determinants of matrices with large integers as entries.
The strangest thing of this bug is that for some matrices, the determinant function can give different values. The Mathematica notebook which reproduces the bug is available here.
