# 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 $1$ – Anurag A Aug 22 '15 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.
$\left( \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{array} \right)$
• 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 '15 at 9:32