Equality of image spaces in Macaulay 2

Question

I am starting to use Macaulay 2 to try to characterize a certain space (related to some of my research). It seems I fail at understanding some basics of the software.

I defined a map f: R^4 -> R^4, where R is a commutative ring. I can find a minimal set of generators using mingens kernel f, which gives me

i49 : mingens kernel f

o49 = {2} | -x -1 xy-z  xz-y  |
      {2} | -y -z -z2+1 0     |
      {2} | x  1  0     0     |
      {2} | z  y  0     -z2+1 |

              4       4
o49 : Matrix R  <--- R

So far, all good, and I can check this with kernel f == image mingens kernel f, which is true. When I'm trying to check it "by hand", it fails:

i53 : image matrix {{-x,-1,x*y-z,x*z-y},{-y,-z,-z^2+1,0},{x,1,0,0},{z,y,0,-z^2+1}} == kernel f

o53 = false

Am I doing something very stupid here?

Thanks a lot for the help!