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Is there a set of matrices that satisfy all of the following constraints?

1) $A^2=0, B^2=0, C^2=0...$ where $A,B,C,D..$ are different matrices.

2) All of them commute.

Edit: 3) $AB \neq 0, AC \neq 0, BC \neq 0, ...$.

  • 1
    What about all entries 0 except an $a$ in the top right corner, $a$ ranging over the elements of your field.2012-10-28
  • 0
    edited. Forgot to mention 3).2012-10-28
  • 2
    My answer to [this earlier question of yours](http://math.stackexchange.com/questions/222040/matrices-that-satisfy-ab-0-and-a2-b2/222042#222042) works here too.2012-10-28

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