Pensieve header: A determinant bug in mathematica version 9. (Fixed in 9.0.1)

$Version

mat is a 2x2 matrix with entries rational functions in x, y, z.

{{a,b}, {c,d}}=mat= {{-1-(z (1-2 x+(x^2)-y+3 x y-2 (x^2) y-x (y^2)+(x^2) (y^2)-z+2 x z-(x^2) z+y z-5 x y z+3 (x^2) y z+2 x (y^2) z-2 (x^2) (y^2) z+x y (z^2)-(x^2) y (z^2)-x (y^2) (z^2)+(x^2) (y^2) (z^2)))/(1-2 x+(x^2)-2 y+4 x y-2 (x^2) y+(y^2)-2 x (y^2)+(x^2) (y^2)-2 z+4 x z-2 (x^2) z+4 y z-10 x y z+6 (x^2) y z-2 (y^2) z+8 x (y^2) z-5 (x^2) (y^2) z-x (y^3) z+(x^2) (y^3) z+(z^2)-2 x (z^2)+(x^2) (z^2)-2 y (z^2)+8 x y (z^2)-5 (x^2) y (z^2)+(y^2) (z^2)-8 x (y^2) (z^2)+6 (x^2) (y^2) (z^2)+2 x (y^3) (z^2)-2 (x^2) (y^3) (z^2)-x y (z^3)+(x^2) y (z^3)+2 x (y^2) (z^3)-2 (x^2) (y^2) (z^3)-x (y^3) (z^3)+(x^2) (y^3) (z^3)),(y (-1+z) (1-2 x+(x^2)+x y-(x^2) y+x z-(x^2) z-3 x y z+2 (x^2) y z+x (y^2) z-(x^2) (y^2) z+x y (z^2)-(x^2) y (z^2)-x (y^2) (z^2)+(x^2) (y^2) (z^2)))/(1-2 x+(x^2)-2 y+4 x y-2 (x^2) y+(y^2)-2 x (y^2)+(x^2) (y^2)-2 z+4 x z-2 (x^2) z+4 y z-10 x y z+6 (x^2) y z-2 (y^2) z+8 x (y^2) z-5 (x^2) (y^2) z-x (y^3) z+(x^2) (y^3) z+(z^2)-2 x (z^2)+(x^2) (z^2)-2 y (z^2)+8 x y (z^2)-5 (x^2) y (z^2)+(y^2) (z^2)-8 x (y^2) (z^2)+6 (x^2) (y^2) (z^2)+2 x (y^3) (z^2)-2 (x^2) (y^3) (z^2)-x y (z^3)+(x^2) y (z^3)+2 x (y^2) (z^3)-2 (x^2) (y^2) (z^3)-x (y^3) (z^3)+(x^2) (y^3) (z^3))},{((-1+y) z (1-2 x+(x^2)-y+2 x y-(x^2) y-z+2 x z-(x^2) z+y z-5 x y z+3 (x^2) y z+x (y^2) z-(x^2) (y^2) z+x y (z^2)-(x^2) y (z^2)-x (y^2) (z^2)+(x^2) (y^2) (z^2)))/(1-2 x+(x^2)-2 y+4 x y-2 (x^2) y+(y^2)-2 x (y^2)+(x^2) (y^2)-2 z+4 x z-2 (x^2) z+4 y z-10 x y z+6 (x^2) y z-2 (y^2) z+8 x (y^2) z-5 (x^2) (y^2) z-x (y^3) z+(x^2) (y^3) z+(z^2)-2 x (z^2)+(x^2) (z^2)-2 y (z^2)+8 x y (z^2)-5 (x^2) y (z^2)+(y^2) (z^2)-8 x (y^2) (z^2)+6 (x^2) (y^2) (z^2)+2 x (y^3) (z^2)-2 (x^2) (y^3) (z^2)-x y (z^3)+(x^2) y (z^3)+2 x (y^2) (z^3)-2 (x^2) (y^2) (z^3)-x (y^3) (z^3)+(x^2) (y^3) (z^3)),-1-(y (1-2 x+(x^2)-y+2 x y-(x^2) y-z+3 x z-2 (x^2) z+y z-5 x y z+3 (x^2) y z+x (y^2) z-(x^2) (y^2) z-x (z^2)+(x^2) (z^2)+2 x y (z^2)-2 (x^2) y (z^2)-x (y^2) (z^2)+(x^2) (y^2) (z^2)))/(1-2 x+(x^2)-2 y+4 x y-2 (x^2) y+(y^2)-2 x (y^2)+(x^2) (y^2)-2 z+4 x z-2 (x^2) z+4 y z-10 x y z+6 (x^2) y z-2 (y^2) z+8 x (y^2) z-5 (x^2) (y^2) z-x (y^3) z+(x^2) (y^3) z+(z^2)-2 x (z^2)+(x^2) (z^2)-2 y (z^2)+8 x y (z^2)-5 (x^2) y (z^2)+(y^2) (z^2)-8 x (y^2) (z^2)+6 (x^2) (y^2) (z^2)+2 x (y^3) (z^2)-2 (x^2) (y^3) (z^2)-x y (z^3)+(x^2) y (z^3)+2 x (y^2) (z^3)-2 (x^2) (y^2) (z^3)-x (y^3) (z^3)+(x^2) (y^3) (z^3))}};

The determinant of mat as computed by Det is not the same as ad-bc:

diff=Simplify[a*d-b*c-Det[mat]]

And the difference is not merely a failure of Simplify; indeed the difference diff is non-zero even when specialized to numerical values of x, y, z:

vals=Thread[{x,y,z} -> N[{(Sqrt[2]), π, E}]]

diff /.vals

(a*d-b*c /. vals)-Det[mat /. vals]