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What does "if the rightmost column of the augmented matrix contains a pivot then the system has no solutions" mean?

I thought the augmented matrix:

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

Has a unique solution. Why does my book say that if the last column contains a pivot then there are no solutions?

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As I interpret it, they mean by "augmented matrix" the full coefficient matrix, where the equation $$ \cases{ax + by = c\\dx+ey=f} $$ gets rewritten to the matrix $$ \begin{bmatrix}a&b&c\\d&e&f\end{bmatrix} $$ I hope you can see that if this matrix has a pivot in the right-most column, then the system has no solution.

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    $\begingroup$ (+1) OH I see. I thought without c and f. Thank you. $\endgroup$
    – M.S.E
    Jun 21, 2015 at 13:49

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