Let $A$ be a $n \times n$ matrix and suppose $A$ has a zero submatrix of order $p \times q$ where $p + q \ge n+1$. Then $\det(A) = 0$.
I can see this happening when doing Laplace expansion. I can also prove it using induction but it fills more than a page and with a lot of if-then-else. Can somebody suggest a short and 'elegant' proof?