Definition of rank of a matrix

Question

Can I define the rank of a matrix(A) as the number of non zero rows in RREF(A)? Here's my reason: Let number of zero rows be $x$
Then these rows are the linearly dependent rows of A and $x=dim(left null space)=m-r$.
So number of non zero rows is equal to $rows-x=m-(m-r)=r$.

Answer 1

Yes, this also follows immediately from the fact that the Gauss-algorithm leaves the rank of a matrix unchanged. Since the rank $\operatorname{RREF}(A)$ is the number of its non-zero rows the claim follows.