# Row Echelon Form with Zero-ed Row

Given that I have an augmented matrix in Row Echelon Form or Reduced Row Echelon form, and the bottom row(s) contain only zeroes.

Generally speaking (disregarding fringe cases if any?):

• Why do rows of only zero imply the presence of a free variable?

• Can you have a free variable without the bottom row(s) containing only zeroes?

• Does the presence of a free variable always mean you have an infinite number of solutions?

• They don't, just means you have a lineraly dependant vector. Yes, if you have more variables than equations. Yes, because its free, it means it can be anything, hence infinite number of soltuons – XRBtoTheMOON Feb 11 '18 at 23:45
• So it's incorrect to say that : if there is a row of zeroes, then there are infinitely many solutions – mathguy Feb 11 '18 at 23:46

Why do rows of only zero imply the presence of a free variable?

This is not true, consider matrices with more rows than columns like $$\left(\begin{matrix}1 & 0 \\ 0 & 1 \\ 0 & 0\end{matrix}\right).$$ Free variables are implied by columns without pivot element.

Can you have a free variable without the bottom row(s) containing only zeroes?

Yes, consider $$\left(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0\end{matrix}\right).$$

Does the presence of a free variable always mean you have an infinite number of solutions?

This depends on the size of the base field. If it is infinite (like $\mathbb{Q}$ or $\mathbb{R}$) then yes. Otherwise no.

• What about this thread math.stackexchange.com/questions/309724/… is the accepted answer saying that a row of zeroes implies the presence of free variables? – mathguy Feb 12 '18 at 0:08
• I don't see an accepted answer there. But as I wrote: Free variables are implied by columns without pivot element. This is the correct characterization. – azimut Feb 12 '18 at 0:11
• Sorry, i meant the top answer, but point taken ty – mathguy Feb 12 '18 at 0:12