Rank of a matrix and its submatrices

  • Thread starter Grothard
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Homework Statement



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.


Homework Equations


Determinant formulas


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.
 

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