Solution
= Solution
Let $U,V$ be $n\times n$ <upper triangular matrices>. For an entry below the diagonal, $i>j$, <matrix multiplication> gives
$$
(UV)_{ij}=\sum_{k=1}^nU_{ik}V_{kj}.
$$
If $k<i$, then $U_{ik}=0$. If $k\geq i$, then $k>j$, so $V_{kj}=0$. Every summand is zero, giving
$$
\boxed{i>j\implies(UV)_{ij}=0.}
$$
Thus $UV$ is an <upper triangular matrix>. Equivalently, a potentially nonzero summand would require $i\leq k\leq j$, impossible below the diagonal.