0005.tex

\begin{exercise}{E2}{Reduced row echelon form}{0005}
\begin{exerciseStatement}

\begin{enumerate}[(a)]
\item Show that \[\operatorname{RREF} \left[\begin{array}{ccccc}
1 & -3 & 7 & -4 & 4 \\
2 & -5 & 11 & -7 & 7 \\
1 & 1 & -5 & 0 & 0
\end{array}\right] = \left[\begin{array}{ccccc}
1 & 0 & -2 & -1 & 1 \\
0 & 1 & -3 & 1 & -1 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] .\]
\item Explain why the matrix \(B= \left[\begin{array}{ccccc}
0 & 0 & 1 & -1 & 0 \\
1 & 4 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] \) is or is not in reduced row echelon form.
\end{enumerate}

    \end{exerciseStatement}
\begin{exerciseAnswer}

\begin{enumerate}[(a)]
\item \(\operatorname{RREF} \left[\begin{array}{ccccc}
1 & -3 & 7 & -4 & 4 \\
2 & -5 & 11 & -7 & 7 \\
1 & 1 & -5 & 0 & 0
\end{array}\right] = \left[\begin{array}{ccccc}
1 & 0 & -2 & -1 & 1 \\
0 & 1 & -3 & 1 & -1 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] .\)
\item \(B\) is not in reduced row echelon form because the pivots are not descending to the right. 
\end{enumerate}

    \end{exerciseAnswer}
\end{exercise}