Given a $n\times n$ matrix M. Prove that M has rank less than $k$ if and only if all of the determinants of its $k\times k$ minors are $0$.
My progress: I have thought about this problem for a while, but I only got that if the $k*k$ minors are $0$, then the $k$ columns of that submatrix are linearly dependent. I don't see how to relate this fact to the linear dependence of the original $k$ columns of matrix M. Can anyone please help me with a detailed explanation?
Hint: use the fact that row and column operations do not change the rank of a matrix. If you have a minor with nonzero determinant, you can use row and column operations to turn it into an identity submatrix.