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<h1 class="heading"><a href="linear-algebra-for-team-based-inquiry-learning.html"><span class="title">Linear Algebra for Team-Based Inquiry Learning</span></a></h1>
<p class="byline">Steven Clontz, Drew Lewis</p>
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<a href="E.html" data-scroll="E"><span class="codenumber">1</span> <span class="title">Systems of Linear Equations (E)</span></a><ul>
<li><a href="E1.html" data-scroll="E1">Linear Systems, Vector Equations, and Augmented Matrices (E1)</a></li>
<li><a href="E2.html" data-scroll="E2" class="active">Row Reduction of Matrices (E2)</a></li>
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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="E2"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">1.2</span> <span class="title">Row Reduction of Matrices (E2)</span>
</h2>
<article class="definition definition-like" id="definition-5"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">1.2.1</span><span class="period">.</span>
</h6>
<p id="p-36">Two systems of linear equations (and their corresponding augmented matrices) are said to be <dfn class="terminology">equivalent</dfn> if they have the same solution set.</p>
<p id="p-37">For example, both of these systems share the same solution set \(\setList{ \left[\begin{array}{c} 1 \\ 1\end{array}\right] }\text{.}\)</p>
<div class="sidebyside"><div class="sbsrow" style="margin-left:0%;margin-right:0%;">
<div class="sbspanel" style="width:50%;justify-content:flex-start;"><div class="displaymath" id="p-38">
\begin{align*}
3x_1 &\,-\,& 2x_2 &\,=\,& 1 \\
x_1 &\,+\,& 4x_2 &\,=\,& 5
\end{align*}
</div></div>
<div class="sbspanel" style="width:50%;justify-content:flex-start;"><div class="displaymath" id="p-39">
\begin{align*}
3x_1 &\,-\,& 2x_2 &\,=\,& 1 \\
4x_1 &\,+\,& 2x_2 &\,=\,& 6
\end{align*}
</div></div>
</div></div>
<p id="p-40">Therefore these augmented matrices are equivalent, which we denote with \(\sim\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cc|c}
3 & -2 & 1\\
1 & 4 & 5\\
\end{array}\right]
\sim
\left[\begin{array}{cc|c}
3 & -2 & 1\\
4 & 2 & 6\\
\end{array}\right]
\end{equation*}
</div></article><article class="activity project-like" id="activity-4"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.1</span><span class="period">.</span>
</h6>
<p id="p-41">Following are seven procedures used to manipulate an augmented matrix. Label the procedures that would result in an equivalent augmented matrix as <em class="emphasis">valid</em>, and label the procedures that might change the solution set of the corresponding linear system as <em class="emphasis">invalid</em>.</p>
<ol class="lower-alpha">
<li id="li-24">Swap two rows.</li>
<li id="li-25">Swap two columns.</li>
<li id="li-26">Add a constant to every term in a row.</li>
<li id="li-27">Multiply a row by a nonzero constant.</li>
<li id="li-28">Add a constant multiple of one row to another row.</li>
<li id="li-29">Replace a column with zeros.</li>
<li id="li-30">Replace a row with zeros.</li>
</ol></article><article class="definition definition-like" id="definition-6"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">1.2.2</span><span class="period">.</span>
</h6>
<p id="p-42">The following <dfn class="terminology">row operations</dfn> produce equivalent augmented matrices:</p>
<ol class="decimal">
<li id="li-31">
<p id="p-43">Swap two rows, for example, \(R_1\leftrightarrow R_2\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cc|c} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]
\sim
\left[\begin{array}{cc|c} 4 & 5 & 6 \\ 1 & 2 & 3 \end{array}\right]
\end{equation*}
</div>
</li>
<li id="li-32">
<p id="p-44">Multiply a row by a nonzero constant, for example, \(2R_1\rightarrow R_1\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cc|c} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]
\sim
\left[\begin{array}{cc|c} 2(1) & 2(2) & 2(3) \\ 4 & 5 & 6 \end{array}\right]
\end{equation*}
</div>
</li>
<li id="li-33">
<p id="p-45">Add a constant multiple of one row to another row, for example, \(R_2-4R_1\rightarrow R_2\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cc|c} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]
\sim
\left[\begin{array}{cc|c} 1 & 2 & 3 \\ 4-4(1) & 5-4(2) & 6-4(3) \end{array}\right]
\end{equation*}
</div>
</li>
</ol>
<p id="p-46">Whenever two matrices \(A,B\) are equivalent (so whenever we do any of these operations), we write \(A\sim B\text{.}\)</p></article><article class="activity project-like" id="activity-5"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.2</span><span class="period">.</span>
</h6>
<p id="p-47">Consider the following (equivalent) linear systems.</p>
<ol class="decimal">
<li id="li-34"><div class="displaymath">
\begin{align*}
x &\,+\,& 2y &\,+\,& z &\,=\,& 3 \\
-x &\,-\,& y &\,+\,& z &\,=\,& 1 \\
2x &\,+\,& 5y &\,+\,& 3z &\,=\,& 7
\end{align*}
</div></li>
<li id="li-35"><div class="displaymath">
\begin{align*}
2x &\,+\,& 5y &\,+\,& 3z &\,=\,& 7 \\
-x &\,-\,& y &\,+\,& z &\,=\,& 1 \\
x &\,+\,& 2y &\,+\,& z &\,=\,& 3
\end{align*}
</div></li>
<li id="li-36"><div class="displaymath">
\begin{align*}
x & & &\,-\,& z &\,=\,& 1 \\
& & y &\,+\,& z &\,=\,& 1 \\
& & y &\,+\,& 2z &\,=\,& 4
\end{align*}
</div></li>
<li id="li-37"><div class="displaymath">
\begin{align*}
x &\,+\,& 2y &\,+\,& z &\,=\,& 3 \\
& & y &\,+\,& z &\,=\,& 1 \\
2x &\,+\,& 5y &\,+\,& 3z &\,=\,& 7
\end{align*}
</div></li>
<li id="li-38"><div class="displaymath">
\begin{align*}
x & & &\,-\,& z &\,=\,& 1 \\
& & y &\,+\,& z &\,=\,& 1 \\
& & & & z &\,=\,& 3
\end{align*}
</div></li>
<li id="li-39"><div class="displaymath">
\begin{align*}
x &\,+\,& 2y &\,+\,& z &\,=\,& 3 \\
& & y &\,+\,& z &\,=\,& 1 \\
& & y &\,+\,& 2z &\,=\,& 4
\end{align*}
</div></li>
</ol>
<p id="p-48">Rank the six linear systems from most complicated to simplest.</p></article><article class="activity project-like" id="activity-6"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.3</span><span class="period">.</span>
</h6>
<p id="p-49">We can rewrite the previous in terms of equivalences of augmented matrices</p>
<div class="displaymath">
\begin{align*}
\left[\begin{array}{ccc|c} 2 & 5 & 1 3 & 7 \\ -1 & -1 & 1 & 1 \\ 1 & 2 & 1 & 3 \end{array}\right] & \sim &
\left[\begin{array}{ccc|c} \circledNumber{1} & 2 & 1 & 3 \\ -1 & -1 & 1 & 1 \\ 2 & 5 & 1 & 3 \end{array}\right] & \sim &
\left[\begin{array}{ccc|c} \circledNumber{1} & 2 & 1 & 3 \\ 0 & 1 & 1 & 1 \\ 2 & 5 & 1 & 3 \end{array}\right] \sim \\
\left[\begin{array}{ccc|c} \circledNumber{1} & 2 & 1 & 3 \\ 0 & \circledNumber{1} & 1 & 1 \\ 0 & 1 & 2 & 4 \end{array}\right] & \sim &
\left[\begin{array}{ccc|c} \circledNumber{1} & 0 & -1 & 1 \\ 0 & \circledNumber{1} & 1 & 1 \\ 0 & 1 & 2 & 4 \end{array}\right] & \sim &
\left[\begin{array}{ccc|c} \circledNumber{1} & 0 & -1 & 1 \\ 0 & \circledNumber{1} & 1 & 1 \\ 0 & 0 & \circledNumber{1} & 3 \end{array}\right]
\end{align*}
</div>
<p id="p-50">Determine the row operation(s) necessary in each step to transform the most complicated system's augmented matrix into the simplest.</p></article><article class="definition definition-like" id="definition-7"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">1.2.3</span><span class="period">.</span>
</h6>
<p id="p-51">A matrix is in <dfn class="terminology">reduced row echelon form</dfn> (<dfn class="terminology">RREF</dfn>) if</p>
<ol class="decimal">
<li id="li-40">The leading term (first nonzero term) of each nonzero row is a 1. Call these terms <dfn class="terminology">pivots</dfn>.</li>
<li id="li-41">Each pivot is to the right of every higher pivot.</li>
<li id="li-42">Each term above or below a pivot is zero.</li>
<li id="li-43">All rows of zeroes are at the bottom of the matrix.</li>
</ol>
<p id="p-52">Every matrix has a unique reduced row echelon form. If \(A\) is a matrix, we write \(\RREF(A)\) for the reduced row echelon form of that matrix.</p></article><article class="activity project-like" id="activity-7"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.4</span><span class="period">.</span>
</h6>
<p id="p-53">Recall that a matrix is in <dfn class="terminology">reduced row echelon form</dfn> (<dfn class="terminology">RREF</dfn>) if</p>
<ol class="decimal">
<li id="li-44">The leading term (first nonzero term) of each nonzero row is a 1. Call these terms <dfn class="terminology">pivots</dfn>.</li>
<li id="li-45">Each pivot is to the right of every higher pivot.</li>
<li id="li-46">Each term above or below a pivot is zero.</li>
<li id="li-47">All rows of zeroes are at the bottom of the matrix.</li>
</ol>
<div class="sidebyside"><div class="sbsrow" style="margin-left:0%;margin-right:0%;"><div class="sbspanel" style="width:100%;justify-content:flex-start;"><ol class="upper-alpha">
<li id="li-48"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c} 1 & 0 & 0 & 3 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]
\end{equation*}
</div></li>
<li id="li-49"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c} 1 & 2 & 4 & 3 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]
\end{equation*}
</div></li>
<li id="li-50"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c} 0 & 0 & 0 & 0 \\ 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & -1 \end{array}\right]
\end{equation*}
</div></li>
<li id="li-51"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c} 1 & 0 & 2 & -3 \\ 0 & 3 & 3 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]
\end{equation*}
</div></li>
<li id="li-52"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c} 0 & 1 & 0 & 7 \\ 1 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 \end{array}\right]
\end{equation*}
</div></li>
<li id="li-53"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 7 \\ 0 & 0 & 1 & 0 \end{array}\right]
\end{equation*}
</div></li>
</ol></div></div></div>
<p id="p-54">For each matrix, circle the leading terms, and label it as RREF or not RREF. For the ones not in RREF, find their RREF.</p></article><article class="remark remark-like" id="remark-5"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">1.2.4</span><span class="period">.</span>
</h6>
<p id="p-55">In practice, if we simply need to convert a matrix into reduced row echelon form, we use technology to do so.</p>
<p id="p-56">However, it is also important to understand the <dfn class="terminology">Gauss-Jordan elimination</dfn> algorithm that a computer or calculator uses to convert a matrix (augmented or not) into reduced row echelon form. Understanding this algorithm will help us better understand how to interpret the results in many applications we use it for in Module V.</p></article><article class="activity project-like" id="activity-8"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.5</span><span class="period">.</span>
</h6>
<p id="p-57">Consider the matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}2 & 6 & -1 & 6 \\ 1 & 3 & -1 & 2 \\ -1 & -3 & 2 & 0 \end{array}\right].
\end{equation*}
</div>
<p data-braille="continuation">Which row operation is the best choice for the first move in converting to RREF?</p>
<ol class="lower-alpha">
<li id="li-54">Add row 3 to row 2 (\(R_2+R_3 \rightarrow R_2\))</li>
<li id="li-55">Add row 2 to row 3 (\(R_3+R_2 \rightarrow R_3\))</li>
<li id="li-56">Swap row 1 to row 2 (\(R_1 \leftrightarrow R_2\))</li>
<li id="li-57">Add -2 row 2 to row 1 (\(R_1-2R_2 \rightarrow R_1\))</li>
</ol></article><article class="activity project-like" id="activity-9"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.6</span><span class="period">.</span>
</h6>
<p id="p-58">Consider the matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc} \circledNumber{1} & 3 & -1 & 2 \\ 2 & 6 & -1 & 6 \\ -1 & -3 & 2 & 0 \end{array}\right].
\end{equation*}
</div>
<p data-braille="continuation">Which row operation is the best choice for the next move in converting to RREF?</p>
<ol class="lower-alpha">
<li id="li-58">Add row 1 to row 3 (\(R_3+R_1 \rightarrow R_3\))</li>
<li id="li-59">Add -2 row 1 to row 2 (\(R_2-2R_1 \rightarrow R_2\))</li>
<li id="li-60">Add 2 row 2 to row 3 (\(R_3+2R_2 \rightarrow R_3\))</li>
<li id="li-61">Add 2 row 3 to row 2 (\(R_2+2R_3 \rightarrow R_2\))</li>
</ol></article><article class="activity project-like" id="activity-10"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.7</span><span class="period">.</span>
</h6>
<p id="p-59">Consider the matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}\circledNumber{1} & 3 & -1 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \end{array}\right].
\end{equation*}
</div>
<p data-braille="continuation">Which row operation is the best choice for the next move in converting to RREF?</p>
<ol class="lower-alpha">
<li id="li-62">Add row 1 to row 2 (\(R_2+R_1 \rightarrow R_2\))</li>
<li id="li-63">Add -1 row 3 to row 2 (\(R_2-R_3 \rightarrow R_2\))</li>
<li id="li-64">Add -1 row 2 to row 3 (\(R_3-R_2 \rightarrow R_3\))</li>
<li id="li-65">Add row 2 to row 1 (\(R_1+R_2 \rightarrow R_1\))</li>
</ol></article><article class="activity project-like" id="activity-11"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.8</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-2">
<p id="p-60">Consider the matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 0 & 0 \\ 3 & -1 & 1 \end{array}\right].
\end{equation*}
</div>
</div>
<article class="task exercise-like" id="task-3"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-61">Perform three row operations to produce a matrix closer to RREF.</p></article><article class="task exercise-like" id="task-4"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-62">Finish putting it in RREF.</p></article></article><article class="activity project-like" id="activity-12"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.9</span><span class="period">.</span>
</h6>
<p id="p-63">Consider the matrix</p>
<div class="displaymath">
\begin{equation*}
A=\left[\begin{array}{cccc}2 & 3 & 2 & 3 \\ -2 & 1 & 6 & 1 \\ -1 & -3 & -4 & 1 \end{array}\right].
\end{equation*}
</div>
<p id="p-64">Compute \(\RREF(A)\text{.}\)</p></article><article class="activity project-like" id="activity-13"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.2.10</span><span class="period">.</span>
</h6>
<p id="p-65">Consider the matrix</p>
<div class="displaymath">
\begin{equation*}
A=\left[\begin{array}{cccc} 2 & 4 & 2 & -4 \\ -2 & -4 & 1 & 1 \\ 3 & 6 & -1 & -4 \end{array}\right].
\end{equation*}
</div>
<p id="p-66">Compute \(\RREF(A)\text{.}\)</p></article><article class="remark remark-like" id="remark-6"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">1.2.5</span><span class="period">.</span>
</h6>
<p id="p-67">A video example of how to perform the Gauss-Jordan Elimination algorithm by hand is available at <a class="external" href="https://youtu.be/Cq0Nxk2dhhU" target="_blank"><code class="code-inline tex2jax_ignore">https://youtu.be/Cq0Nxk2dhhU</code></a>.</p>
<p id="p-68">Practicing several exercises on your own using this method is strongly recommended.</p></article><section class="exercises" id="exercises-2"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">1.2.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-6"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-66">Show that<div class="displaymath">
\begin{equation*}
\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] .
\end{equation*}
</div>
</li>
<li id="li-67">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.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-6" id="answer-6"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-6"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-68">\(\displaystyle \operatorname{RREF} \left[\begin{array}{ccccc}
1 & -2 & -1 & 3 & 4 \\
2 & -3 & -3 & 5 & 6 \\
0 & -5 & 5 & 5 & 10
\end{array}\right] = \left[\begin{array}{ccccc}
1 & 0 & -3 & 1 & 0 \\
0 & 1 & -1 & -1 & -2 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] .\)</li>
<li id="li-69">\(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-7"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-70">Show that<div class="displaymath">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
0 & 0 & 1 & -2 \\
1 & -3 & 5 & -12 \\
0 & 0 & -4 & 8 \\
1 & -3 & 3 & -8
\end{array}\right] = \left[\begin{array}{cccc}
1 & -3 & 0 & -2 \\
0 & 0 & 1 & -2 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] .
\end{equation*}
</div>
</li>
<li id="li-71">Explain why the matrix \(B= \left[\begin{array}{cccc}
1 & 0 & -1 & -3 \\
-7 & 1 & 5 & 20 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \) is or is not in reduced row echelon form.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-7" id="answer-7"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-7"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-72">\(\displaystyle \operatorname{RREF} \left[\begin{array}{cccc}
0 & 0 & 1 & -2 \\
1 & -3 & 5 & -12 \\
0 & 0 & -4 & 8 \\
1 & -3 & 3 & -8
\end{array}\right] = \left[\begin{array}{cccc}
1 & -3 & 0 & -2 \\
0 & 0 & 1 & -2 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] .\)</li>
<li id="li-73">\(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-8"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-74">Show that<div class="displaymath">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccc}
3 & -10 & 0 \\
-2 & 7 & 0 \\
1 & 2 & 0 \\
2 & -7 & 0 \\
3 & -9 & 0
\end{array}\right] = \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] .
\end{equation*}
</div>
</li>
<li id="li-75">Explain why the matrix \(B= \left[\begin{array}{ccc}
1 & 0 & -2 \\
-2 & 1 & 4 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] \) is or is not in reduced row echelon form.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-8" id="answer-8"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-8"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-76">\(\displaystyle \operatorname{RREF} \left[\begin{array}{ccc}
3 & -10 & 0 \\
-2 & 7 & 0 \\
1 & 2 & 0 \\
2 & -7 & 0 \\
3 & -9 & 0
\end{array}\right] = \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] .\)</li>
<li id="li-77">\(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-9"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-78">Show that<div class="displaymath">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
1 & -1 & -1 & -8 \\
2 & -1 & -5 & -12 \\
0 & -3 & 9 & -11 \\
-1 & 1 & 1 & 7
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & -4 & 0 \\
0 & 1 & -3 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right] .
\end{equation*}
</div>
</li>
<li id="li-79">Explain why the matrix \(B= \left[\begin{array}{cccc}
1 & 0 & 0 & -3 \\
0 & 0 & 1 & 3 \\
0 & 1 & 0 & -3 \\
0 & 0 & 0 & 0
\end{array}\right] \) is or is not in reduced row echelon form.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-9" id="answer-9"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-9"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-80">\(\displaystyle \operatorname{RREF} \left[\begin{array}{cccc}
1 & -1 & -1 & -8 \\
2 & -1 & -5 & -12 \\
0 & -3 & 9 & -11 \\
-1 & 1 & 1 & 7
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & -4 & 0 \\
0 & 1 & -3 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right] .\)</li>
<li id="li-81">\(B\) is not in reduced row echelon form because the pivots are not descending to the right.</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-10"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-82">Show that<div class="displaymath">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccc}
2 & -2 & 0 \\
-5 & 1 & 4 \\
-2 & 3 & -1 \\
-5 & -2 & 7 \\
1 & -2 & 1
\end{array}\right] = \left[\begin{array}{ccc}
1 & 0 & -1 \\
0 & 1 & -1 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] .
\end{equation*}
</div>
</li>
<li id="li-83">Explain why the matrix \(B= \left[\begin{array}{ccc}
0 & 1 & -2 \\
1 & 0 & 2 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] \) is or is not in reduced row echelon form.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-10" id="answer-10"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-10"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-84">\(\displaystyle \operatorname{RREF} \left[\begin{array}{ccc}
2 & -2 & 0 \\
-5 & 1 & 4 \\
-2 & 3 & -1 \\
-5 & -2 & 7 \\
1 & -2 & 1
\end{array}\right] = \left[\begin{array}{ccc}
1 & 0 & -1 \\
0 & 1 & -1 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] .\)</li>
<li id="li-85">\(B\) is not in reduced row echelon form because the pivots are not descending to the right.</li>
</ol></div></div>
</div></article><p id="p-69"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/E2/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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