Past exam of the mathematical methods I course of the University of Oxford / 2019 resit / 4 / Solution

Use the Leibniz formula for determinants:For a nonzero product in an upper triangular matrix, every selected column must satisfy $\sigma (j)\ge j$. But a permutation has ${\sum }_j\sigma (j)={\sum }_{j}j$, so every one of these inequalities is an equality. Only the identity permutation can contribute. Its sign is positive and its product is ${\prod }_j{m}_{jj}$, which proves the result even if some diagonal entries vanish.