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I want to make a set of matrices that satisfies all the following:

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

2) $AB = 0$, $CD = 0$.....

3) All matrices in the set commute.

4) $AC \neq 0$, $AD \neq 0$, $BC \neq 0$, $BD \neq 0$.... so other than $AB$, $CD$ ... multiplication is always nonzero.

Also, is there any mathematical object that satisfies all of these?

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    By better, I mean that the size of matrices being shorter.2012-10-27

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There's at least $ A=B=\begin{bmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{bmatrix} \qquad C=D=\begin{bmatrix}0&0&1&0\\0&0&0&1\\0&0&0&0\\0&0&0&0\end{bmatrix}$ I think this ought to generalize to $n$ pairs of $2^n\times 2^n$ matrices. Each matrix is a Kronecker product of $n$ matrices of which one is $\begin{bmatrix}0&1\\0&0\end{bmatrix}$ and the rest are $2\times 2$ identity matrices.

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.