The row rank and column rank of a matrix are the same

Let A be a matrix over a field. Prove that the row rank and column rank of A are equal.


Let A be an arbitrary matrix, and let C be the reduced row echelon form of A. Suppose P is the invertible matrix such that PA = C. We showed in two previous exercises (here and here) that A and C have the same row rank and the same column rank. We showed here that the row rank and column rank of C are equal. Thus the row rank and column rank of A are equal.

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