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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?

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

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

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