# Reduced-row echelon form associated to three lines in the plane

Let $\ell_1,\ell_2$ and $\ell_3$ be three lines in the plane $\mathbb{R}^2$. For $i = 1, 2, 3$, let the line $\ell_i$ have equation $a_i x + b_i y = c_i$.

Is it possible for the matrix $$\left(\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{array}\right)$$ to have reduced row echelon form equal to $$\left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array}\right) ?$$

• Not possible because the third column is for the third variable (say) $z$, which in your case happens to be $1$ Aug 22, 2015 at 9:31

If you extract row 1 from the matrix you have $0x + 0y = 1$ which is not possible, meaning the system of line equations is inconsistent. Geometrically that means that the three lines do not have a common intersection point.

The three lines you described in the comment below have matrix form:

$\left( \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{array} \right)$

• The three lines having respective equation x = 0, y = 0, x + y = 0 have no common intersection yet the reduced row echelon form has first row (1 0 0), second row (0 1 0) and third row (0 0 1).
– user17982
Aug 23, 2015 at 6:58
• It is possible that the system is inconsistent yet the reduced row echelon form is not equal to the matrix with first row (0 0 1), second row (0 0 0) and third row (0 0 0).
– user17982
Aug 23, 2015 at 9:22
• if any rows have of an augmented matrix which is in reduced echelon form are all zeros except for the last column, then the solution is inconsistent. Think about what that is saying. That row would be $0x + 0y = 1$ which is impossible, therefore the system is inconsistent.
– Jack
Aug 23, 2015 at 9:32
• Agreed, the system is inconsistent. Still, that does not rule out that the reduced row echelon form has first row (0 0 1) and all zeroes for the second and third rows.
– user17982
Aug 24, 2015 at 11:37