0009.tex

\begin{exercise}{E2}{Reduced row echelon form}{0009}
\begin{exerciseStatement}

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

    \end{exerciseStatement}
\begin{exerciseAnswer}

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

    \end{exerciseAnswer}
\end{exercise}