- #1

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## Homework Statement

Let ##n \in \mathbb{Z}^+## and let ##F## be a field. Prove that the set ##H = \{(A_{ij}) \in GL_n (F) ~ | ~ A_{ij} = 0 ~ \forall i > j \}## is a subgroup of ##GL_n (F)##

## Homework Equations

## The Attempt at a Solution

So clearly the set is nonempty since ##I_n## is upper triangular. Now let ##A, B \in H##. We want to show that ##(AB)_{ij} = 0 ~ \forall i > j##. So ##(AB)_{ij} = \sum_{k=1}^n A_{ik}B_{kj}##. Now here is where I'm a bit stuck. I know that the fact that ##A## and ##B## are both in ##H## means that when ##i > j## we'll have ##\sum_{k=1}^n A_{ik}B_{kj} = 0##. But I am not sure how to show this clearly...