# Rank of a matrix and its submatrices

by Grothard
Tags: linear
 P: 30 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.

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