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Are there a set of different $n$ matrices that commute that

1) after $n$ multiplications of these matrices - that is for example, $A \times B \times C \times ...A_n$, where $\times$ represents matrix multiplication, if the multiplied result was a product of some different matrices and two or more equal matrices, the result of multiplication is triangular. Otherwise, the result is not triangular.

Does this set exist for all numbers for $n$ bigger than 4?

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    What do you mean by "if the matrix can be decomposed into some matrices and two or more equal matrices"?2012-10-29
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    Edited. hope this clears.2012-10-29
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    I think you mean you are looking for fixed positive integers $n$ and $m$ and a set $S$ consisting of $n$ distinct $m \times m$ matrices such that for every product $P = A_1 \ldots A_n$ with all $A_j \in S$ and at least two of the $A_j$ are equal, $P$ is triangular, but if all $A_j$ are distinct then $P$ is not triangular. Is that right?2012-10-29
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    That's correct.2012-10-29

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