Past exam of the mathematical methods I course of the University of Oxford / 2021 / 3 / a / Solution

In the matrix product, $(AB{)}_{ij}={\sum }_{k=1}^n{a}_{ik}{b}_{kj}$. A nonzero summand in a product of upper triangular matrices requires both $i\le k$ and $k\le j$. If $i>j$, no such index exists, so the product is upper triangular. If $i=j$, only $k=i$ survives, givingThis also shows inductively why the diagonal of an integer matrix power consists of the corresponding powers of the diagonal entries.