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Q. True or False: If matrix A is a reduced row-echelon form, then at least one of the entries in each column must be 1.

It comes down to this question.

Can I have the following as Reduced Row-Echelon form? $$\left(\begin{array}{cccc} 0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0 \end{array}\;\begin{array}{c}\end{array}\right)$$

If that is reduced row echelon form, that is false right? Please let me know. Thanks!

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Yes, you are right....imo at least, as this could depend on definitions. – DonAntonio Apr 25 '13 at 3:02
Sure, yours is a good counterexample. One could give an example that involves less typing, like the $1\times 1$ zero matrix! Maybe pedagogically better is the $2\times 2$, first row $1\quad0$, second row $0\quad 0$. – André Nicolas Apr 25 '13 at 3:16
up vote 1 down vote accepted

Here's another example that answers the question: $\left(\begin{smallmatrix} 1&0&0\\0&0&1\\0&0&0\end{smallmatrix}\right)$.

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