 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2019-resit/4/solution

det M = \sum_{σ \in S_{n}} sgn (σ) \prod_{j = 1}^{n} m_{j, σ (j)} .

For a nonzero product in an upper triangular matrix, every selected column must satisfy

σ (j) \geq j

\sum_{j} σ (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.

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