\begin{exercise}{E2}{Reduced row echelon form}{0000}
\begin{exerciseStatement}
\begin{enumerate}[(a)]
\item Show that \[\operatorname{RREF} \left[\begin{array}{ccccc}
1 & -2 & -1 & 3 & 4 \\
2 & -3 & -3 & 5 & 6 \\
0 & -5 & 5 & 5 & 10
\end{array}\right] = \left[\begin{array}{ccccc}
1 & 0 & -3 & 1 & 0 \\
0 & 1 & -1 & -1 & -2 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] .\]
\item Explain why the matrix \(B= \left[\begin{array}{ccccc}
1 & 4 & 0 & -1 & 1 \\
-5 & -20 & 1 & 7 & -6 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] \) is or is not in reduced row echelon form.
\end{enumerate}
\end{exerciseStatement}
\begin{exerciseAnswer}
\begin{enumerate}[(a)]
\item \(\operatorname{RREF} \left[\begin{array}{ccccc}
1 & -2 & -1 & 3 & 4 \\
2 & -3 & -3 & 5 & 6 \\
0 & -5 & 5 & 5 & 10
\end{array}\right] = \left[\begin{array}{ccccc}
1 & 0 & -3 & 1 & 0 \\
0 & 1 & -1 & -1 & -2 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] .\)
\item \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.
\end{enumerate}
\end{exerciseAnswer}
\end{exercise}
