Register to reply

Rank of a matrix and its submatrices

by Grothard
Tags: linear
Share this thread:
Nov11-11, 03:58 PM
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.
Phys.Org News Partner Science news on
An interesting glimpse into how future state-of-the-art electronics might work
Tissue regeneration using anti-inflammatory nanomolecules
C2D2 fighting corrosion

Register to reply

Related Discussions
Rank of matrix Calculus & Beyond Homework 0
Matrix Multiplication and Rank of Matrix General Math 2
Rank of a matrix Linear & Abstract Algebra 5
Matrix manipulations/rank of a matrix Calculus & Beyond Homework 2
Rank of a Matrix Calculus & Beyond Homework 6