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Here is the proof that I am trying to show.

Let A be an m x n matrix in RREF. If rank(A) = r < m, then A must have at least one row of zeroes.

So far, I've noticed that is true, and that if for example, there are 5 equations, but only 4 leading ones, then the last row must all be zeroes. I'm just having trouble on how to write up the formal proof of this.

Thanks

Ian Murphy
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$rank(A) = r < m$ means that the number of pivots in the elimination process is less than $m$. This directly indicates that at $m - r$ rows are linearly dependent and therefore will become zero after elimination. Since the matrix is in $rref$ form then it is already undergone the elimination.

user1
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