Triangular Matrices forms Subring of Square Matrices
Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\map {\MM_R} n$ be the order $n$ square matrix space over a ring $R$.
Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$.
Let $\map {U_R} n$ be the set of upper triangular matrices of order $n$ over $R$.
Then $\map {U_R} n$ forms a subring of $\struct {\map {\MM_R} n, +, \times}$.
Similarly, let $\map {L_R} n$ be the set of lower triangular matrices of order $n$ over $R$.
Then $\map {L_R} n$ forms a subring of $\struct {\map {\MM_R} n, +, \times}$.
Proof
From Negative of Triangular Matrix, if $\mathbf B \in \map {U_R} n$ then $-\mathbf B \in \map {U_R} n$.
Then from Sum of Triangular Matrices, if $\mathbf A, -\mathbf B \in \map {U_R} n$ then $\mathbf A + \paren {-\mathbf B} \in \map {U_R} n$.
From Product of Triangular Matrices, if $\mathbf A, \mathbf B \in \map {U_R} n$ then $\mathbf A \mathbf B \in \map {U_R} n$.
The result follows from the Subring Test.
The same argument can be applied to matrices in $\map {L_R} n$.
$\blacksquare$
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.1$: Subrings: Examples $3$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $4$ (limited result)