Let $\operatorname{Rank}(A)=k$. Why must some $k$-by-$k$ submatrix of $A$ have nonzero determinant?

Question

Let $\operatorname{Rank}(A)=k$. Why must some $k$-by-$k$ submatrix of $A$ have nonzero determinant? And why does every $(k+1)$-by-$(k+1)$ submatrix of $A$ have zero determinant?

Answer 1

Since the rank of $A$ is $k$, there are $k$ linearly independent rows; remove the others, so you get a $k\times n$ matrix having rank $k$. Since this matrix has $k$ linearly independent columns, remove the others.

The matrix you get is $k\times k$ and has rank $k$.

A $(k+1)\times(k+1)$ submatrix can't have rank $k+1$, otherwise also the original matrix would.