I'd appreciate any help with this one:
Let K be a field, $n,k\geq 1$ and $A \in M(n,n;K) $. Show:
$rank(A)\geq k \Leftrightarrow \exists i_1,...,i_k \in \{1,...,n\}, j_1,...,j_k\in\{1,...,n\}$ so that the matrix $B= \Biggl(\begin{smallmatrix} A_{i_1 j_1}& ...&A_{i_1 j_k} \\ \vdots&\ddots&\vdots \\ A_{i_k j_1} & ...&A_{i_k j_k } \end{smallmatrix} \Biggr)$ is invertible.
For $k=n$ of course it's trivial. But after that I'm stuck.
Thanks in advance! QC