Rank of a matrix and its submatrices

Grothard

1. The problem statement, all variables and given/known data

Let A be a nonzero matrix of size n. Let a k*k submatrix of A be defined as a matrix obtained by deleting any n-k rows and n-k columns of A. Let m denote the largest integer such that some m*m submatrix has a nonzero determinant. Prove that rank(A) = k.

Now conversely suppose that rank(A) = m. Prove that some m*m submatrix has a nonzero determinant.


2. Relevant equations
Determinant formulas


3. The attempt at a solution

Not quite sure if I should proceed by examining the solution space of A or rather just do something clever with the determinants. I feel like there's a property of determinants that I'm missing that'd make this much easier.