# Why can Echelon Matrices have zero rows but Echelon systems can't have any equations with no leading variables?

According to the definition my professor gave us its okay for a matrix in echelon form to have a zero row, but a system of equations in echelon form cannot have an equation with no leading variable.

Why is this? Aren't they supposed to represent the same thing?

$$\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}$$

• I don't get it. – user98295 Oct 2 '13 at 20:48
• What's a “leading variable”? – Michael Hoppe Oct 2 '13 at 21:01
• The non-zero variable furthest to the left in a row. – user98295 Oct 2 '13 at 21:02

I think the distinction you're referring to is the difference between the echelon forms of the coefficient matrix of a system versus the augmented matrix of the system. Suppose you have the system

$ax_1+bx_2=c$

$dx_1+ex_2=f.$

Then there is the coefficient matrix $\bigl( \begin{smallmatrix} a & b \\ d & e \\ \end{smallmatrix} \bigr)$

and there is the augmented matrix $\bigl( \begin{smallmatrix} a & b & c \\ d & e & f \\ \end{smallmatrix} \bigr)$.

It may happen that the row reduced echelon form of the coefficient matrix has a zero row, e.g. $\bigl( \begin{smallmatrix} \ast & \ast \\ 0 & 0 \\ \end{smallmatrix} \bigr)$. However, if that row is not a zero row in the row reduced echelon form for the augmented matrix, then the rref of the augmented matrix looks like $\bigl( \begin{smallmatrix} \ast & \ast & \ast \\ 0 & 0 & 1 \end{smallmatrix} \bigr)$, and that last row corresponds to the equation

$0x_1+0x_2=1$ which is clearly a contradiction, hence the original system has no solutions.