illusiia

2021-02-14

Prove that if A has rank r with r > 0, then A has an r×r invertible submatrix.

sweererlirumeX

Skilled2021-02-15Added 91 answers

It is given that rank of A is r. Then A has rr independent columns. Let B be the submatrix formed by those rr columns of A. Clearly the rank of B is r. We know that rank of a matrix is equal to dimension of the row space of that matrix. Therefore dimension of the row space of B also r. Therese rr rows are linearly independemmt. Form the submatrix B if we choose particularly those r rows, then is an r×r submatrix which is invertible.

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