The product of upper/lower triangular matrices is upper/lower triangular.

Theorem. The product of upper triangular matrices is upper triangular, and the product of lower triangular matrices is lower triangular.

If $A$ and $B$ are both $n \times n$ upper triangular matrices, then $AB$ is also upper triangular. Similarly, if $A$ and $B$ are lower triangular, then $AB$ is lower triangular.

Proof idea. For upper triangular matrices, if $i > j$, then $(AB)_{ij} = \sum_{k=1}^n a_{ik} b_{kj}$. Since $a_{ik} = 0$ when $k < i$ and $b_{kj} = 0$ when $k > j$, and $i > j$, every term in the sum is zero.

Example.
$$\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 4 & 5 \\ 0 & 6 \end{pmatrix} = \begin{pmatrix} 4 & 17 \\ 0 & 18 \end{pmatrix}$$