## 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.