---

Example

The augmented matrix

\[\begin{equation*}
  \begin{pmatrix}
    1 & 5 & 2 & -6 \\
    0 & 4 & -7 & 2 \\
    0 & 0 & 0 & 5
  \end{pmatrix}
\end{equation*}
\]

represents an inconsistent system, for per the last row

\[\begin{align*}
  (x_1, x_2, x_3) & = (0, 0, 0) \\
   x_1 + x_2 + x_3 & = 5,
\end{align*}
\]

which is impossible, as

\[\begin{equation*}
  0 + 0 + 0 \neq 5.
\end{equation*}
\]