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Prove that if $A$ and $B$ are both upper or lower triangular matrices, then the diagonal entries of both $AB$ and $BA$ are the products of the diagonal entries.

Attempt. Assuming the dimensions of matrices A and B allow their product to be successfully computed. I don't know how to go about this proof after this step.

Hint. I want the solution to this problem to be a simple as possible, while using techniques that are elementary by nature. So please, no definition of Matrix Multiplication or Summations. Thanks!

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  • $\begingroup$ I think you should change upper and lower triangular matrices to upper or lower triangular matrices. $\endgroup$ – TheGeekGreek Sep 24 '16 at 21:34
  • $\begingroup$ Done @TheGeekGreek $\endgroup$ – Jose Dos Santos Sep 24 '16 at 21:35
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Hint. Try to use the multiplication formula for matrices (if they are of appropriate sizes) $$(AB)_{ij} = \sum_{k = 1}^n A_{ik}B_{kj}$$ and consider the case where $j = i$ to get the diagonal entries.

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  • $\begingroup$ Is there an easier or less tedious way to do this than to use summations $\endgroup$ – Jose Dos Santos Sep 24 '16 at 21:34
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    $\begingroup$ I really believe this is the easiest way to do the job. You only want diagonals, so the sum tells you what to do. $\endgroup$ – Sean Roberson Sep 24 '16 at 21:36
  • $\begingroup$ @JoseDosSantos Yes, I think it is the easiest one. A less tedious way would just be to say, that this is trivial. $\endgroup$ – TheGeekGreek Sep 24 '16 at 21:38

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