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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">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="M2"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">4.2</span> <span class="title">Row Operations as Matrix Multiplication (M2)</span>
</h2>
<article class="activity project-like" id="activity-111"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">4.2.1</span><span class="period">.</span>
</h6>
<p id="p-721">Let \(A=\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]\text{.}\) Find a \(3 \times 3\) matrix \(B\) such that \(BA=A\text{,}\) that is,</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc} \unknown & \unknown & \unknown \\
\unknown & \unknown & \unknown
\\ \unknown & \unknown & \unknown \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">Check your guess using technology.</p></article><article class="definition definition-like" id="definition-25"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">4.2.1</span><span class="period">.</span>
</h6>
<p id="p-722">The identity matrix \(I_n\) (or just \(I\) when \(n\) is obvious from context) is the \(n \times n\) matrix</p>
<div class="displaymath">
\begin{equation*}
I_n = \left[\begin{array}{cccc} 1 & 0 & \cdots & 0 \\ 0 & 1 & \ddots & \vdots \\
\vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right].
\end{equation*}
</div>
<p data-braille="continuation">It has a \(1\) on each diagonal element and a \(0\) in every other position.</p></article><article class="fact theorem-like" id="fact-21"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">4.2.2</span><span class="period">.</span>
</h6>
<p id="p-723">For any square matrix \(A\text{,}\) \(IA=AI=A\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
\end{equation*}
</div></article><article class="activity project-like" id="activity-112"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">4.2.2</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-34"><p id="p-724">Tweaking the identity matrix slightly allows us to write row operations in terms of matrix multiplication.</p></div>
<article class="task exercise-like" id="task-81"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-725">Create a matrix that doubles the third row of \(A\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 2 & 2 & -2 \end{array}\right]
\end{equation*}
</div></article><article class="task exercise-like" id="task-82"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-726">Create a matrix that swaps the second and third rows of \(A\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 1 & 1 & -1 \\ 0 & 3 & 2 \end{array}\right]
\end{equation*}
</div></article><article class="task exercise-like" id="task-83"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-727">Create a matrix that adds \(5\) times the third row of \(A\) to the first row:</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2+5(1) & 7+5(1) & -1+5(-1) \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
\end{equation*}
</div></article></article><article class="fact theorem-like" id="fact-22"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">4.2.3</span><span class="period">.</span>
</h6>
<p id="p-728">If \(R\) is the result of applying a row operation to \(I\text{,}\) then \(RA\) is the result of applying the same row operation to \(A\text{.}\)</p>
<ul class="disc">
<li id="li-521"><p id="p-729">Scaling a row: \(R=
\left[\begin{array}{ccc}
c & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\)</p></li>
<li id="li-522"><p id="p-730">Swapping rows: \(R=
\left[\begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right]\)</p></li>
<li id="li-523"><p id="p-731">Adding a row multiple to another row: \(R=
\left[\begin{array}{ccc}
1 & 0 & c \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\)</p></li>
</ul>
<p id="p-732">Such matrices can be chained together to emulate multiple row operations. In particular,</p>
<div class="displaymath">
\begin{equation*}
\RREF(A)=R_k\dots R_2R_1A
\end{equation*}
</div>
<p data-braille="continuation">for some sequence of matrices \(R_1,R_2,\dots,R_k\text{.}\)</p></article><article class="activity project-like" id="activity-113"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">4.2.3</span><span class="period">.</span>
</h6>
<p id="p-733">Consider the two row operations \(R_2\leftrightarrow R_3\) and \(R_1+R_2\to R_1\) applied as follows to show \(A\sim B\text{:}\)</p>
<div class="displaymath">
\begin{align*}
A
=
\left[\begin{array}{ccc}
-1&4&5\\
0&3&-1\\
1&2&3\\
\end{array}\right]
&\sim
\left[\begin{array}{ccc}
-1&4&5\\
1&2&3\\
0&3&-1\\
\end{array}\right]\\
&\sim
\left[\begin{array}{ccc}
-1+1&4+2&5+3\\
1&2&3\\
0&3&-1\\
\end{array}\right]
=
\left[\begin{array}{ccc}
0&6&8\\
1&2&3\\
0&3&-1\\
\end{array}\right]
=
B
\end{align*}
</div>
<p id="p-734">Express these row operations as matrix multiplication by expressing \(B\) as the product of two matrices and \(A\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
B =
\left[\begin{array}{ccc}
\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown
\end{array}\right]
\left[\begin{array}{ccc}
\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown
\end{array}\right]
A
\end{equation*}
</div>
<p data-braille="continuation">Check your work using technology.</p></article><section class="exercises" id="exercises-18"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">4.2.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-86"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-735">Let \(A\) be a \(4 \times 4\) matrix.</p>
<ol class="lower-alpha">
<li id="li-524">Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \(R_1 \to R_1 + 2R_4 \text{.}\)</li>
<li id="li-525">Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \(R_1 \to -2R_1 \text{.}\)</li>
<li id="li-526">Use matrix multiplication to describe the matrix obtained by applying \(R_1 \to R_1 + 2R_4 \) and then \(R_1 \to -2R_1 \) to \(A\) (note the order).</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-86" id="answer-86"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-86"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-527">\(\displaystyle B= \left[\begin{array}{cccc}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \)</li>
<li id="li-528">\(\displaystyle C= \left[\begin{array}{cccc}
-2 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \)</li>
<li id="li-529">\(\displaystyle CBA\)</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-87"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-736">Let \(A\) be a \(4 \times 4\) matrix.</p>
<ol class="lower-alpha">
<li id="li-530">Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \(R_3 \to 3R_3 \text{.}\)</li>
<li id="li-531">Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \(R_4 \leftrightarrow R_1 \text{.}\)</li>
<li id="li-532">Use matrix multiplication to describe the matrix obtained by applying \(R_3 \to 3R_3 \) and then \(R_4 \leftrightarrow R_1 \) to \(A\) (note the order).</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-87" id="answer-87"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-87"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-533">\(\displaystyle C= \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 3 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \)</li>
<li id="li-534">\(\displaystyle N= \left[\begin{array}{cccc}
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0
\end{array}\right] \)</li>
<li id="li-535">\(\displaystyle NCA\)</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-88"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-737">Let \(A\) be a \(4 \times 4\) matrix.</p>
<ol class="lower-alpha">
<li id="li-536">Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \(R_2 \to R_2 + 5R_3 \text{.}\)</li>
<li id="li-537">Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \(R_4 \to -2R_4 \text{.}\)</li>
<li id="li-538">Use matrix multiplication to describe the matrix obtained by applying \(R_4 \to -2R_4 \) and then \(R_2 \to R_2 + 5R_3 \) to \(A\) (note the order).</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-88" id="answer-88"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-88"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-539">\(\displaystyle B= \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 5 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \)</li>
<li id="li-540">\(\displaystyle C= \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -2
\end{array}\right] \)</li>
<li id="li-541">\(\displaystyle BCA\)</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-89"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-738">Let \(A\) be a \(4 \times 4\) matrix.</p>
<ol class="lower-alpha">
<li id="li-542">Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \(R_4 \to 5R_4 \text{.}\)</li>
<li id="li-543">Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \(R_2 \to R_2 + 2R_3 \text{.}\)</li>
<li id="li-544">Use matrix multiplication to describe the matrix obtained by applying \(R_2 \to R_2 + 2R_3 \) and then \(R_4 \to 5R_4 \) to \(A\) (note the order).</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-89" id="answer-89"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-89"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-545">\(\displaystyle M= \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 5
\end{array}\right] \)</li>
<li id="li-546">\(\displaystyle C= \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 2 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \)</li>
<li id="li-547">\(\displaystyle MCA\)</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-90"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-739">Let \(A\) be a \(4 \times 4\) matrix.</p>
<ol class="lower-alpha">
<li id="li-548">Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \(R_1 \to R_1 + 2R_2 \text{.}\)</li>
<li id="li-549">Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \(R_3 \leftrightarrow R_2 \text{.}\)</li>
<li id="li-550">Use matrix multiplication to describe the matrix obtained by applying \(R_3 \leftrightarrow R_2 \) and then \(R_1 \to R_1 + 2R_2 \) to \(A\) (note the order).</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-90" id="answer-90"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-90"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-551">\(\displaystyle Q= \left[\begin{array}{cccc}
1 & 2 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \)</li>
<li id="li-552">\(\displaystyle C= \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \)</li>
<li id="li-553">\(\displaystyle QCA\)</li>
</ol></div></div>
</div></article><p id="p-740"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/M2/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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