How to prove the product of upper triangular matrices is upper triangular?

[Brucezhou]

This seems easy but when I tried to do this, the best way I came up with is to list all entries and then do the multiplication work. Is there any better ,clearer and more simple way to do the proof?

 

[SteamKing]

This seems easy but when I tried to do this, the best way I came up with is to list all entries and then do the multiplication work. Is there any better ,clearer and more simple way to do the proof?

Did you try to express the entries in the product matrix in terms of the dot products of the row of one matrix with the corresponding column in the second matrix?

 

[vela]

Suppose A is an upper triangular matrix with elements $a_{ij}$. Then you know that $a_{ij}=0$ if $i>j$. If C=AB where B is also upper triangular, you want to show that $c_{ij}=0$ if $i>j$.