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

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.

For the upper triangular matrix $A=\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right)$, direct matrix multiplication givesThe real diagonal entries are therefore $a=3,d=4$. Their off-diagonal multiplier is $9+12+16=37$, so $b=-74/37=-2$. A required matrix isIt is the unique real upper triangular cube root. Allowing complex diagonal cube roots produces other complex matrices; uniqueness is asserted only over the real numbers.