Product of atomic triangular matrices

Hello everybody,

I am trying to prove the following: I am having a triangular matrix (with ), I can create atomic triangular matrices (in other words: For I set all elements that are not on the main diagonal and that are not in column to be zero.) Then I need to show that this is true:

So far I have found: (what might even be the wrong attempt)

where: is an Element of the result after I performed the first multiplications. and is an Element of the atomoc triangular matrix .

However, this is where I am stuck: I cannot figure out a way how to prove by the last equation that all the elements must be as in the original matrix.

Thank you for ideas / suggestions / solutions!

Best, Rafael