Triangular Matrices forms Subring of Square Matrices

Theorem
Let $$\mathcal M_K \left({n}\right)$$ be the order $n$ square matrix space over a field $$K$$.

Let $$U_K \left({n}\right)$$ be the set of upper triangular matrices of order $$n$$ over $$K$$.

Then $$U_K \left({n}\right)$$ forms a subring of $$\mathcal M_K \left({n}\right)$$ on the operations of matrix entrywise addition and conventional matrix multiplication.

Similarly, let $$L_K \left({n}\right)$$ be the set of lower triangular matrices of order $$n$$ over $$K$$.

Then $$L_K \left({n}\right)$$ forms a subring of $$\mathcal M_K \left({n}\right)$$ on the operations of matrix entrywise addition and conventional matrix multiplication.

Proof
From Negative of Triangular Matrix, if $$\mathbf B \in U_K \left({n}\right)$$ then $$-\mathbf B \in U_K \left({n}\right)$$.

Then from Sum of Triangular Matrices, if $$\mathbf A, -\mathbf B \in U_K \left({n}\right)$$ then $$\mathbf A + \left({-\mathbf B}\right) \in U_K \left({n}\right)$$.

From Product of Triangular Matrices, if $$\mathbf A, \mathbf B \in U_K \left({n}\right)$$ then $$\mathbf A \mathbf B \in U_K \left({n}\right)$$.

The result follows from the Subring Test.

The same argument can be applied to matrices in $$L_K \left({n}\right)$$.