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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) $AB \neq 0, AC \neq 0, BC \neq 0, ...$.

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1  
What about all entries 0 except an $a$ in the top right corner, $a$ ranging over the elements of your field. –  Jason Polak Oct 28 '12 at 0:26
    
edited. Forgot to mention 3). –  TTTY Oct 28 '12 at 0:47
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My answer to this earlier question of yours works here too. –  Henning Makholm Oct 28 '12 at 1:05

1 Answer 1

Let $$E=\left(\begin{array}{cc} 0 & 1\\ 0 & 0 \end{array}\right), \ \ \ \ I=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right).$$

Then set

$$ A = E\otimes I\otimes I\otimes I,\\ B = I\otimes E\otimes I\otimes I,\\ C = I\otimes I\otimes E\otimes I,\\ D = I\otimes I\otimes I\otimes E. $$

Where $\otimes$ is the Kronecker product. These were constructed following the procedure outline by @HenningMakholm here:

More generally if only you have an $A$ and $B$ with $A^2=B^2$, $AB=BA=0$, you can combine $n$ copies of this structure using Kronecker products. That automatically makes matrices from different pairs commute, and ensures that cross-level products are nonzero as long as the generators are.

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