Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Digital systems, linear algebra, introduction to formal proof / 3 / a / Solution

Let $U,V$ be $n\times n$ upper triangular matrices. For an entry below the diagonal, $i>j$, matrix multiplication givesIf $k<i$, then $U_{ik}=0$. If $k\ge i$, then $k>j$, so $V_{kj}=0$. Every summand is zero, givingThus $UV$ is an upper triangular matrix. Equivalently, a potentially nonzero summand would require $i\le k\le j$, impossible below the diagonal.