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

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.