IMC 2001 · Problem 4

Let $A = (a_{k,\ell})_{k,\ell = 1,\dots,n}$ be an $n \times n$ complex matrix such that for each $m \in \{1, \dots, n\}$ and $1 \le j_1 < \dots < j_m \le n$ the determinant of the matrix

$$(a_{j_k, j_\ell})_{k,\ell = 1,\dots,m}$$

is zero. Prove that $A^n = 0$ and that there exists a permutation $\sigma \in S_n$ such that the matrix

$$(a_{\sigma(k), \sigma(\ell)})_{k,\ell = 1,\dots,n}$$

has all of its nonzero elements above the diagonal.