Bug as of Ruby 1.6.x--1.8.x (has been fixed in trunk as of revision 5599
without explicit mention as far as I can tell):

Maxima 5.14.0 
(%i25) determinant(matrix([2,1],[1,-1]));
(%o25)                                - 3

Mathematica 6.0
In[1]:= Det[{{2, 1}, {1, -1}}]
Out[1]= -3

Cramer's rule:
2*(-1) - 1*1 = -3

But Ruby's matrix.rb says:

>> Matrix[[2,1],[1,-1]].det
=> -2

-- 
Christian Neukirchen  <chneukirchen / gmail.com>  http://chneukirchen.org