# 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 Feb 11, 2018 at 23:45
• So it's incorrect to say that : if there is a row of zeroes, then there are infinitely many solutions Feb 11, 2018 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.