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Consider the set $V$ of upper-triangular $n\times n$ matrices with elements in some field $K$. I.e., if $A$ is such a matrix, $a_{ij} = 0$, for $i > j$

Show that non-degenerate upper-triangular matrices form a group with respect to matrix multiplication.

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Well, if $A$ is upper-triangular, then $A^{-1}$ is upper-triangular as well, and if $A,B$ are upper-triangular, then $AB$ is, too. This follows from the formulas for matrix multiplication. For different proofs of this see here. Since the identity is upper-triangular, the invertible upper-triangular matrices form a group, e.g., a subgroup of $GL(n,K)$.

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