Suppose that there is a set of matrices, and these matrices commute.
When matrices are multiplied, they are not triangular, but when the multiplication of matrices are multiplied with one of its matrices - that is if the multiplication is ABCD, then multiply with either A, B, C or D - it becomes triangular. (It does not matter whether the original matrices are triangular or not.)
So are there any matrices that satisfy these properties?
Edit: So, this is what I want exactly:
1) Suppose there are $n$ matrices. These $n$ matrices may be triangular or not triangular.
2) We do any $n$ multiplications - that is $ABCD....A_n$ (any combination - it might be $AC...A_n$ and so on). When there contains any two matrices that are same, it becomes triangular. Otherwise, it's not triangular.
3) The question also, is this set of matrices possible for all numbers for $n$?