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<!DOCTYPE html>
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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>
<li><a href="E3.html" data-scroll="E3">Solving Linear Systems (E3)</a></li>
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<a href="V.html" data-scroll="V"><span class="codenumber">2</span> <span class="title">Vector Spaces (V)</span></a><ul>
<li><a href="V1.html" data-scroll="V1">Vector Spaces (V1)</a></li>
<li><a href="V2.html" data-scroll="V2">Linear Combinations (V2)</a></li>
<li><a href="V3.html" data-scroll="V3">Spanning Sets (V3)</a></li>
<li><a href="V4.html" data-scroll="V4">Subspaces (V4)</a></li>
<li><a href="V5.html" data-scroll="V5">Linear Independence (V5)</a></li>
<li><a href="V6.html" data-scroll="V6">Identifying a Basis (V6)</a></li>
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<li><a href="A2.html" data-scroll="A2">Standard Matrices (A2)</a></li>
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<li><a href="A4.html" data-scroll="A4">Injective and Surjective Linear Maps (A4)</a></li>
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<a href="G.html" data-scroll="G"><span class="codenumber">5</span> <span class="title">Geometric Properties of Linear Maps (G)</span></a><ul>
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<li><a href="G2.html" data-scroll="G2">Computing Determinants (G2)</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="G1"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">5.1</span> <span class="title">Row Operations and Determinants (G1)</span>
</h2>
<article class="activity project-like" id="activity-120"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.1</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-37">
<p id="p-787">The image below illustrates how the linear transformation \(T : \IR^2 \rightarrow \IR^2\) given by the standard matrix \(A = \left[\begin{array}{cc} 2 & 0 \\ 0 & 3 \end{array}\right]\) transforms the unit square.</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.5]
\fill[red!50!white] (0,0) rectangle (1,1);`
\draw[thin,gray,<->] (-4,0)-- (4,0);
\draw[thin,gray,<->] (0,-4)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(A \vec{e}_1= \left[\begin{array}{c}2 \\ 0 \end{array}\right]\)}++ (2,0);
\draw[thick,blue,->] (0,0) -- node[left] {\(A \vec{e}_2 = \left[\begin{array}{c} 0 \\ 3 \end{array}\right]\)}++(0,3);
\draw[thick,red,->] (0,0) -- ++(1,0);
\draw[thick,red,->] (0,0) -- ++(0,1);
\draw[blue,dashed] (2,0) -- (2,3) -- (0,3);
\draw[red,dashed] (1,0) -- (1,1) -- (0,1);
\end{tikzpicture}
</pre>
</div>
<article class="task exercise-like" id="task-90"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-788">What are the lengths of \(A\vec e_1\) and \(A\vec e_2\text{?}\)</p></article><article class="task exercise-like" id="task-91"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-789">What is the area of the transformed unit square?</p></article></article><article class="activity project-like" id="activity-121"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.2</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-38">
<p id="p-790">The image below illustrates how the linear transformation \(S : \IR^2 \rightarrow \IR^2\) given by the standard matrix \(B = \left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\) transforms the unit square.</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.5]
\fill[red!50!white] (0,0) rectangle (1,1);
\draw[thin,gray,<->] (-4,0)-- (4,0);
\draw[thin,gray,<->] (0,-4)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(B \vec{e}_1= \left[\begin{array}{c}2 \\ 0 \end{array}\right]\)}++ (2,0);
\draw[thick,blue,->] (0,0) -- ++(3,4) node[above] {\(B \vec{e}_2 = \left[\begin{array}{c} 3 \\ 4 \end{array}\right]\)};
\draw[thick,red,->] (0,0) -- ++(1,0);
\draw[thick,red,->] (0,0) -- ++(0,1);
\draw[blue,dashed] (2,0) -- (5,4) -- (3,4);
\draw[red,dashed] (1,0) -- (1,1) -- (0,1);
\end{tikzpicture}
</pre>
</div>
<article class="task exercise-like" id="task-92"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-791">What are the lengths of \(B\vec e_1\) and \(B\vec e_2\text{?}\)</p></article><article class="task exercise-like" id="task-93"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-792">What is the area of the transformed unit square?</p></article></article><article class="observation remark-like" id="observation-26"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">5.1.1</span><span class="period">.</span>
</h6>
<p id="p-793">It is possible to find two nonparallel vectors that are scaled but not rotated by the linear map given by \(B\text{.}\)</p>
<div class="displaymath">
\begin{equation*}
B\vec e_1=\left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\left[\begin{array}{c}1\\0\end{array}\right]
=\left[\begin{array}{c}2\\0\end{array}\right]=2\vec e_1
\end{equation*}
</div>
<div class="displaymath">
\begin{equation*}
B\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]
=
\left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]
=
\left[\begin{array}{c}3\\2\end{array}\right]
=
4\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]
\end{equation*}
</div>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.5]
\fill[red!50!white] (0,0) -- (1,0) -- (1.75,0.5) -- (0.75,0.5) -- (0,0);
\draw[thin,gray,<->] (-4,0)-- (4,0);
\draw[thin,gray,<->] (0,-4)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(B\left[\begin{array}{c}1\\0\end{array}\right]=2\left[\begin{array}{c}1\\0\end{array}\right]\)}++ (2,0);
\draw[thick,blue,->] (0,0) -- ++(3,2) node[above] {\(B\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]=4\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]\)};
\draw[thick,red,->] (0,0) -- (1,0);
\draw[thick,red,->] (0,0) -- (0.75,0.5);
\draw[red,dashed] (1,0) -- (1.75,0.5) -- (0.75,0.5);
\draw[blue,dashed] (2,0) -- (5,2) -- (3,2);
\end{tikzpicture}
</pre>
<p id="p-794">The process for finding such vectors will be covered later in this module.</p></article><article class="observation remark-like" id="observation-27"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">5.1.2</span><span class="period">.</span>
</h6>
<p id="p-795">Notice that while a linear map can transform vectors in various ways, linear maps always transform parallelograms into parallelograms, and these areas are always transformed by the same factor: in the case of \(B=\left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\text{,}\) this factor is \(8\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.5]
\fill[red!50!white] (0,0) rectangle (1,1);
\draw[thin,gray,<->] (-4,0)-- (4,0);
\draw[thin,gray,<->] (0,-4)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(B \vec{e}_1= \left[\begin{array}{c}2 \\ 0 \end{array}\right]\)}++ (2,0);
\draw[thick,blue,->] (0,0) -- ++(3,4) node[above] {\(B \vec{e}_2 = \left[\begin{array}{c} 3 \\ 4 \end{array}\right]\)};
\draw[thick,red,->] (0,0) -- ++(1,0);
\draw[thick,red,->] (0,0) -- ++(0,1);
\draw[blue,dashed] (2,0) -- (5,4) -- (3,4);
\draw[red,dashed] (1,0) -- (1,1) -- (0,1);
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\fill[red!50!white] (0,0) -- (1,0) -- (1.75,0.5) -- (0.75,0.5) -- (0,0);
\draw[thin,gray,<->] (-4,0)-- (4,0);
\draw[thin,gray,<->] (0,-4)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(B\left[\begin{array}{c}1\\0\end{array}\right]=2\left[\begin{array}{c}1\\0\end{array}\right]\)}++ (2,0);
\draw[thick,blue,->] (0,0) -- ++(3,2) node[above] {\(B\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]=4\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]\)};
\draw[thick,red,->] (0,0) -- (1,0);
\draw[thick,red,->] (0,0) -- (0.75,0.5);
\draw[red,dashed] (1,0) -- (1.75,0.5) -- (0.75,0.5);
\draw[blue,dashed] (2,0) -- (5,2) -- (3,2);
\end{tikzpicture}
</pre>
<p id="p-796">Since this change in area is always the same for a given linear map, it will be equal to the value of the transformed unit square (which begins with area \(1\)).</p></article><article class="remark remark-like" id="remark-13"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">5.1.3</span><span class="period">.</span>
</h6>
<p id="p-797">We will define the <dfn class="terminology">determinant</dfn> of a square matrix \(A\text{,}\) or \(\det(A)\) for short, to be the factor by which \(A\) scales areas. In order to figure out how to compute it, we first figure out the properties it must satisfy.</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.5]
\fill[red!50!white] (0,0) rectangle (1,1);
\draw[thin,gray,<->] (-4,0)-- (4,0);
\draw[thin,gray,<->] (0,-4)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(B \vec{e}_1= \left[\begin{array}{c}2 \\ 0 \end{array}\right]\)}++ (2,0);
\draw[thick,blue,->] (0,0) -- ++(3,4) node[above] {\(B \vec{e}_2 = \left[\begin{array}{c} 3 \\ 4 \end{array}\right]\)};
\draw[thick,red,->] (0,0) -- ++(1,0);
\draw[thick,red,->] (0,0) -- ++(0,1);
\draw[blue,dashed] (2,0) -- (5,4) -- (3,4);
\draw[red,dashed] (1,0) -- (1,1) -- (0,1);
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\fill[red!50!white] (0,0) -- (1,0) -- (1.75,0.5) -- (0.75,0.5) -- (0,0);
\draw[thin,gray,<->] (-4,0)-- (4,0);
\draw[thin,gray,<->] (0,-4)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(B\left[\begin{array}{c}1\\0\end{array}\right]=2\left[\begin{array}{c}1\\0\end{array}\right]\)}++ (2,0);
\draw[thick,blue,->] (0,0) -- ++(3,2) node[above] {\(B\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]=4\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]\)};
\draw[thick,red,->] (0,0) -- (1,0);
\draw[thick,red,->] (0,0) -- (0.75,0.5);
\draw[red,dashed] (1,0) -- (1.75,0.5) -- (0.75,0.5);
\draw[blue,dashed] (2,0) -- (5,2) -- (3,2);
\end{tikzpicture}
</pre></article><article class="activity project-like" id="activity-122"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.3</span><span class="period">.</span>
</h6>
<p id="p-798">The transformation of the unit square by the standard matrix \([\vec{e}_1\hspace{0.5em} \vec{e}_2]=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]=I\) is illustrated below. What is \(\det([\vec{e}_1\hspace{0.5em} \vec{e}_2])=\det(I)\text{,}\) the area of the transformed unit square shown here?</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=1]
\fill[red!50!white] (0,0) rectangle (1,1);
\draw[thin,gray,<->] (-1,0)-- (3,0);
\draw[thin,gray,<->] (0,-1)-- (0,3);
\draw[thick,blue,->] (0,0) -- node[below] {\(\vec{e}_1=\left[\begin{array}{c}1 \\ 0 \end{array}\right]\)} (1,0);
\draw[thick,blue,->] (0,0) -- node[left] {\(\vec{e}_2=\left[\begin{array}{c} 0 \\ 1 \end{array}\right]\)} (0,1);
\draw[dashed,blue] (1,0) -- (1,1);
\draw[dashed,blue] (0,1) -- (1,1);
\end{tikzpicture}
</pre>
<ol class="lower-alpha">
<li id="li-566"><p id="p-799">0</p></li>
<li id="li-567"><p id="p-800">1</p></li>
<li id="li-568"><p id="p-801">2</p></li>
<li id="li-569"><p id="p-802">4</p></li>
</ol></article><article class="activity project-like" id="activity-123"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.4</span><span class="period">.</span>
</h6>
<p id="p-803">The transformation of the unit square by the standard matrix \([\vec{v}\hspace{0.5em} \vec{v}]\) is illustrated below: both \(T(\vec{e}_1)=T(\vec{e}_2)=\vec{v}\text{.}\) What is \(\det([\vec{v}\hspace{0.5em} \vec{v}])\text{,}\) the area of the transformed unit square shown here?</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.8]
\fill[red!50!white] (0,0) rectangle (1,1);
\draw[thin,gray,<->] (-1,0)-- (4,0);
\draw[thin,gray,<->] (0,-1)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(\vec{v}\)} (3,2);
\end{tikzpicture}
</pre>
<ol class="lower-alpha">
<li id="li-570"><p id="p-804">0</p></li>
<li id="li-571"><p id="p-805">1</p></li>
<li id="li-572"><p id="p-806">2</p></li>
<li id="li-573"><p id="p-807">4</p></li>
</ol></article><article class="activity project-like" id="activity-124"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.5</span><span class="period">.</span>
</h6>
<p id="p-808">The transformations of the unit square by the standard matrices \([\vec{v}\hspace{0.5em} \vec{w}]\) and \([c\vec{v}\hspace{0.5em} \vec{w}]\) are illustrated below. Describe the value of \(\det([c\vec{v}\hspace{0.5em} \vec{w}])\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.8]
\draw[thin,gray,<->] (-1,0)-- (6,0);
\draw[thin,gray,<->] (0,-1)-- (0,6);
\draw[thick,red,->] (0,0) -- node[below right] {\(c\vec{v}\)} (4,2);
\draw[thick,red,->] (1,3) -- (5,5);
\draw[thick,blue,->] (0,0) -- node[below] {\(\vec{v}\)} (3,1.5);
\draw[thick,purple,->] (0,0) -- node[left] {\(\vec{w}\)} (1,3);
\draw[lightgray,very thick,dashed] (1,3) -- (2,1);
\draw[thick,blue,->] (3,1.5) -- (4,4.5);
\draw[thick,blue,->] (1,3) -- (4,4.5);
\draw[thick,red,->] (4,2) -- (5,5);
\end{tikzpicture}
</pre>
<ol class="lower-alpha">
<li id="li-574"><p id="p-809">\(\displaystyle \det([\vec{v}\hspace{0.5em} \vec{w}])\)</p></li>
<li id="li-575"><p id="p-810">\(\displaystyle \det([\vec{v}\hspace{0.5em} \vec{w}])+c\det([\vec{v}\hspace{0.5em} \vec{w}]\)</p></li>
<li id="li-576"><p id="p-811">\(\displaystyle c\det([\vec{v}\hspace{0.5em} \vec{w}])\)</p></li>
<li id="li-577"><p id="p-812">Cannot be determined from this information.</p></li>
</ol></article><article class="activity project-like" id="activity-125"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.6</span><span class="period">.</span>
</h6>
<p id="p-813">The transformations of unit squares by the standard matrices \([\vec{u}\hspace{0.5em} \vec{w}]\text{,}\) \([\vec{v}\hspace{0.5em} \vec{w}]\) and \([\vec{u}+\vec{v}\hspace{0.5em} \vec{w}]\) are illustrated below. Describe the value of \(\det([\vec{u}+\vec{v}\hspace{0.5em} \vec{w}])\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.8]
\draw[thin,gray,<->] (-1,0)-- (6,0);
\draw[thin,gray,<->] (0,-1)-- (0,6);
\draw[thick,blue,->] (0,0) -- node[below] {\(\vec{u}\)} (2,1);
\draw[thick,purple,->] (0,0) -- node[left] {\(\vec{w}\)} (1,3);
\draw[thick,blue,->] (2,1) -- node [below] {\(\vec{v}\)}(3,2);
\draw[thick,purple,->] (2,1) -- (3,4);
\draw[thick,purple,->] (3,2) -- (4,5);
\draw[thick,blue,->] (1,3) -- (3,4);
\draw[thick,blue,->] (3,4) -- (4,5);
\draw[thick,red,->] (0,0) -- (3,2)node[above,right] {\(\vec{u}+\vec{v}\)};
\draw[thick,red,->] (1,3) -- (4,5);
\end{tikzpicture}
</pre>
<ol class="lower-alpha">
<li id="li-578"><p id="p-814">\(\displaystyle \det([\vec{u}\hspace{0.5em} \vec{w}])=\det([\vec{v}\hspace{0.5em} \vec{w}])\)</p></li>
<li id="li-579"><p id="p-815">\(\displaystyle \det([\vec{u}\hspace{0.5em} \vec{w}])+\det([\vec{v}\hspace{0.5em} \vec{w}])\)</p></li>
<li id="li-580"><p id="p-816">\(\displaystyle \det([\vec{u}\hspace{0.5em} \vec{w}])\det([\vec{v}\hspace{0.5em} \vec{w}])\)</p></li>
<li id="li-581"><p id="p-817">Cannot be determined from this information.</p></li>
</ol></article><article class="definition definition-like" id="definition-27"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">5.1.4</span><span class="period">.</span>
</h6>
<p id="p-818">The <dfn class="terminology">determinant</dfn> is the unique function \(\det:M_{n,n}\to\IR\) satisfying these properties:</p>
<ol class="decimal">
<li id="li-582"><p id="p-819">[P1:] \(\det(I)=1\)</p></li>
<li id="li-583"><p id="p-820">[P2:] \(\det(A)=0\) whenever two columns of the matrix are identical.</p></li>
<li id="li-584"><p id="p-821">[P3:] \(\det[\cdots\hspace{0.5em}c\vec{v}\hspace{0.5em}\cdots]=
c\det[\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}\cdots]\text{,}\) assuming no other columns change.</p></li>
<li id="li-585"><p id="p-822">[P4:] \(\det[\cdots\hspace{0.5em}\vec{v}+\vec{w}\hspace{0.5em}\cdots]=
\det[\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}\cdots]+
\det[\cdots\hspace{0.5em}\vec{w}\hspace{0.5em}\cdots]\text{,}\) assuming no other columns change.</p></li>
</ol>
<p id="p-823">Note that these last two properties together can be phrased as ``The determinant is linear in each column.''</p></article><article class="observation remark-like" id="observation-28"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">5.1.5</span><span class="period">.</span>
</h6>
<p id="p-824">The determinant must also satisfy other properties. Consider \(\det([\vec v \hspace{1em}\vec w+c \vec{v}])\) and \(\det([\vec v\hspace{1em}\vec w])\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.5]
\draw[thin,gray,<->] (-1,0)-- (8,0);
\draw[thin,gray,<->] (0,-1)-- (0,6);
\draw[very thick,blue,->] (0,0) -- node[below right] {\(\vec{v}\)} (3,1);
\draw[very thick,blue,->] (0,0) -- node[left] {\(\vec{w}\)} (1,2);
\draw[dashed,blue,->] (1,2) -- (4,3);
\draw[dashed,blue,->] (3,1) -- (4,3);
\draw[thick,red,->] (0,0) -- (5.5,3.5) node[above] {\(\vec{w}+c\vec{v}\)};
\draw[dashed,red,->] (3,1) -- (8.5,4.5);
\draw[dashed,blue,->] (5.5,3.5) -- (8.5,4.5);
\draw[thin,dashed,gray] (3,1) -- (2.5,2.5);
\draw[thin,dashed,gray] (4,3) -- (5.5,3.5);
\end{tikzpicture}
</pre>
<p id="p-825">The base of both parallelograms is \(\vec{v}\text{,}\) while the height has not changed, so the determinant does not change either. This can also be proven using the other properties of the determinant:</p>
<div class="displaymath">
\begin{align*}
\det([\vec{v}+c\vec{w}\hspace{1em}\vec{w}])
&=
\det([\vec{v}\hspace{1em}\vec{w}])+
\det([c\vec{w}\hspace{1em}\vec{w}])\\
&=
\det([\vec{v}\hspace{1em}\vec{w}])+
c\det([\vec{w}\hspace{1em}\vec{w}])\\
&=
\det([\vec{v}\hspace{1em}\vec{w}])+
c\cdot 0\\
&=
\det([\vec{v}\hspace{1em}\vec{w}])
\end{align*}
</div></article><article class="remark remark-like" id="remark-14"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">5.1.6</span><span class="period">.</span>
</h6>
<p id="p-826">Swapping columns may be thought of as a reflection, which is represented by a negative determinant. For example, the following matrices transform the unit square into the same parallelogram, but the second matrix reflects its orientation.</p>
<div class="displaymath">
\begin{equation*}
A=\left[\begin{array}{cc}2&3\\0&4\end{array}\right]\hspace{1em}\det A=8\hspace{3em}
B=\left[\begin{array}{cc}3&2\\4&0\end{array}\right]\hspace{1em}\det B=-8
\end{equation*}
</div>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.5]
\fill[red!50!white] (0,0) rectangle (1,1);
\draw[thin,gray,<->] (-4,0)-- (4,0);
\draw[thin,gray,<->] (0,-4)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(A \vec{e}_1= \left[\begin{array}{c}2 \\ 0 \end{array}\right]\)}++ (2,0);
\draw[thick,blue,->] (0,0) -- ++(3,4) node[above] {\(A \vec{e}_2 = \left[\begin{array}{c} 3 \\ 4 \end{array}\right]\)};
\draw[thick,red,->] (0,0) -- ++(1,0);
\draw[thick,red,->] (0,0) -- ++(0,1);
\draw[blue,dashed] (2,0) -- (5,4) -- (3,4);
\draw[red,dashed] (1,0) -- (1,1) -- (0,1);
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\fill[red!50!white] (0,0) rectangle (1,1);
\draw[thin,gray,<->] (-4,0)-- (4,0);
\draw[thin,gray,<->] (0,-4)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below] {\(B \vec{e}_2= \left[\begin{array}{c}2 \\ 0 \end{array}\right]\)}++ (2,0);
\draw[thick,blue,->] (0,0) -- ++(3,4) node[above] {\(B \vec{e}_1 = \left[\begin{array}{c} 3 \\ 4 \end{array}\right]\)};
\draw[thick,red,->] (0,0) -- ++(1,0);
\draw[thick,red,->] (0,0) -- ++(0,1);
\draw[blue,dashed] (2,0) -- (5,4) -- (3,4);
\draw[red,dashed] (1,0) -- (1,1) -- (0,1);
\end{tikzpicture}
</pre></article><article class="observation remark-like" id="observation-29"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">5.1.7</span><span class="period">.</span>
</h6>
<p id="p-827">The fact that swapping columns multiplies determinants by a negative may be verified by adding and subtracting columns.</p>
<div class="displaymath">
\begin{align*}
\det([\vec{v}\hspace{1em}\vec{w}])
&=
\det([\vec{v}+\vec{w}\hspace{1em}\vec{w}])\\
&=
\det([\vec{v}+\vec{w}\hspace{1em}\vec{w}-(\vec{v}+\vec{w})])\\
&=
\det([\vec{v}+\vec{w}\hspace{1em}-\vec{v}])\\
&=
\det([\vec{v}+\vec{w}-\vec{v}\hspace{1em}-\vec{v}])\\
&=
\det([\vec{w}\hspace{1em}-\vec{v}])\\
&=
-\det([\vec{w}\hspace{1em}\vec{v}])
\end{align*}
</div></article><article class="fact theorem-like" id="fact-23"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">5.1.8</span><span class="period">.</span>
</h6>
<p id="p-828">To summarize, we've shown that the column versions of the three row-reducing operations a matrix may be used to simplify a determinant in the following way:</p>
<ol class="lower-alpha">
<li id="li-586">
<p id="p-829">Multiplying a column by a scalar multiplies the determinant by that scalar:</p>
<div class="displaymath">
\begin{equation*}
c\det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em} \cdots])=
\det([\cdots\hspace{0.5em}c\vec{v}\hspace{0.5em} \cdots])
\end{equation*}
</div>
</li>
<li id="li-587">
<p id="p-830">Swapping two columns changes the sign of the determinant:</p>
<div class="displaymath">
\begin{equation*}
\det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}
\cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])=
-\det([\cdots\hspace{0.5em}\vec{w}\hspace{0.5em}
\cdots\hspace{1em}\vec{v}\hspace{0.5em} \cdots])
\end{equation*}
</div>
</li>
<li id="li-588">
<p id="p-831">Adding a multiple of a column to another column does not change the determinant:</p>
<div class="displaymath">
\begin{equation*}
\det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}
\cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])=
\det([\cdots\hspace{0.5em}\vec{v}+c\vec{w}\hspace{0.5em}
\cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])
\end{equation*}
</div>
</li>
</ol></article><article class="activity project-like" id="activity-126"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.7</span><span class="period">.</span>
</h6>
<p id="p-832">The transformation given by the standard matrix \(A\) scales areas by \(4\text{,}\) and the transformation given by the standard matrix \(B\) scales areas by \(3\text{.}\) By what factor does the transformation given by the standard matrix \(AB\) scale areas?</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[x=0.2in,y=0.2in]
\begin{scope}
\fill[red] (0,0) -- (1,0) -- (1,1) -- (0,1) -- (0,0);
\end{scope}
\draw[->,thick] (2,1) to[bend left=30] node[above] {\(B\)} (3,1);
\begin{scope}[shift={(3,-1)}]
\fill[purple] (0,0) -- (2,1) -- (3,3) -- (1,2) -- (0,0);
\end{scope}
\draw[->,thick] (6.5,1) to[bend left=30] node[above] {\(A\)} (7.5,1);
\begin{scope}[shift={(12,-3)}]
\fill[blue] (0,0) -- (-4,5) -- (-1,7) -- (3,2) -- (0,0);
\end{scope}
\end{tikzpicture}
</pre>
<ol class="lower-alpha">
<li id="li-589"><p id="p-833">\(\displaystyle 1\)</p></li>
<li id="li-590"><p id="p-834">\(\displaystyle 7\)</p></li>
<li id="li-591"><p id="p-835">\(\displaystyle 12\)</p></li>
<li id="li-592"><p id="p-836">Cannot be determined</p></li>
</ol></article><article class="fact theorem-like" id="fact-24"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">5.1.9</span><span class="period">.</span>
</h6>
<p id="p-837">Since the transformation given by the standard matrix \(AB\) is obtained by applying the transformations given by \(A\) and \(B\text{,}\) it follows that</p>
<div class="displaymath">
\begin{equation*}
\det(AB)=\det(A)\det(B)=\det(B)\det(A)=\det(BA)\text{.}
\end{equation*}
</div></article><article class="remark remark-like" id="remark-15"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">5.1.10</span><span class="period">.</span>
</h6>
<p id="p-838">Recall that row operations may be produced by matrix multiplication.</p>
<ul class="disc">
<li id="li-593"><p id="p-839">Multiply the first row of \(A\) by \(c\text{:}\) \(\left[\begin{array}{cccc}
c & 0 & 0 & 0\\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]A\)</p></li>
<li id="li-594"><p id="p-840">Swap the first and second row of \(A\text{:}\) \(\left[\begin{array}{cccc}
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]A\)</p></li>
<li id="li-595"><p id="p-841">Add \(c\) times the third row to the first row of \(A\text{:}\) \(\left[\begin{array}{cccc}
1 & 0 & c & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]A\)</p></li>
</ul></article><article class="fact theorem-like" id="fact-25"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">5.1.11</span><span class="period">.</span>
</h6>
<p id="p-842">The determinants of row operation matrices may be computed by manipulating columns to reduce each matrix to the identity:</p>
<ul class="disc">
<li id="li-596"><p id="p-843">Scaling a row: \(\det
\left[\begin{array}{cccc}
c & 0 & 0 & 0\\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
=
c\det
\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
=
c\)</p></li>
<li id="li-597"><p id="p-844">Swapping rows: \(\det
\left[\begin{array}{cccc}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
=
-1\det
\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
=
-1\)</p></li>
<li id="li-598"><p id="p-845">Adding a row multiple to another row: \(\det
\left[\begin{array}{cccc}
1 & 0 & c & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
=
\det
\left[\begin{array}{cccc}
1 & 0 & c-1c & 0\\
0 & 1 & 0-0c & 0\\
0 & 0 & 1-0c & 0 \\
0 & 0 & 0-0c & 1
\end{array}\right]
=
\det(I)=1\)</p></li>
</ul></article><article class="activity project-like" id="activity-127"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.8</span><span class="period">.</span>
</h6>
<p id="p-846">Consider the row operation \(R_1+4R_3\to R_1\) applied as follows to show \(A\sim B\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
A=\left[\begin{array}{cccc}1&2&3 & 4\\5&6 & 7 & 8\\9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16\end{array}\right]
\sim
\left[\begin{array}{cccc}1+4(9)&2+4(10)&3+4(11) & 4+4(12) \\5&6 & 7 & 8\\9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16\end{array}\right]=B
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-599"><p id="p-847">Find a matrix \(R\) such that \(B=RA\text{,}\) by applying the same row operation to \(I=\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right]\text{.}\)</p></li>
<li id="li-600"><p id="p-848">Find \(\det R\) by comparing with the previous slide.</p></li>
<li id="li-601">
<p id="p-849">If \(C \in M_{3,3}\) is a matrix with \(\det(C)= -3\text{,}\) find</p>
<div class="displaymath">
\begin{equation*}
\det(RC)=\det(R)\det(C).
\end{equation*}
</div>
</li>
</ol></article><article class="activity project-like" id="activity-128"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.9</span><span class="period">.</span>
</h6>
<p id="p-850">Consider the row operation \(R_1\leftrightarrow R_3\) applied as follows to show \(A\sim B\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
A=\left[\begin{array}{cccc}1&2&3&4\\5&6&7&8\\9&10&11&12 \\ 13 & 14 & 15 & 16\end{array}\right]
\sim
\left[\begin{array}{cccc}9&10&11&12\\5&6&7&8\\1&2&3&4 \\ 13 & 14 & 15 & 16\end{array}\right]=B
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-602"><p id="p-851">Find a matrix \(R\) such that \(B=RA\text{,}\) by applying the same row operation to \(I\text{.}\)</p></li>
<li id="li-603"><p id="p-852">If \(C \in M_{3,3}\) is a matrix with \(\det(C)= 5\text{,}\) find \(\det(RC)\text{.}\)</p></li>
</ol></article><article class="activity project-like" id="activity-129"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.1.10</span><span class="period">.</span>
</h6>
<p id="p-853">Consider the row operation \(3R_2\to R_2\) applied as follows to show \(A\sim B\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
A=\left[\begin{array}{cccc}1&2&3&4\\5&6&7&8\\9&10&11&12 \\ 13 & 14 & 15 & 16\end{array}\right]
\sim
\left[\begin{array}{cccc}1&2&3&4\\3(5)&3(6)&3(7)&3(8)\\9&10&11&12 \\ 13 & 14 & 15 & 16\end{array}\right]=B
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-604"><p id="p-854">Find a matrix \(R\) such that \(B=RA\text{.}\)</p></li>
<li id="li-605"><p id="p-855">If \(C \in M_{3,3}\) is a matrix with \(\det(C)= -7\text{,}\) find \(\det(RC)\text{.}\)</p></li>
</ol></article><article class="remark remark-like" id="remark-16"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">5.1.12</span><span class="period">.</span>
</h6>
<p id="p-856">Recall that the column versions of the three row-reducing operations a matrix may be used to simplify a determinant:</p>
<ol class="lower-alpha">
<li id="li-606">
<p id="p-857">Multiplying columns by scalars:</p>
<div class="displaymath">
\begin{equation*}
\det([\cdots\hspace{0.5em}c\vec{v}\hspace{0.5em} \cdots])=
c\det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em} \cdots])
\end{equation*}
</div>
</li>
<li id="li-607">
<p id="p-858">Swapping two columns:</p>
<div class="displaymath">
\begin{equation*}
\det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}
\cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])=
-\det([\cdots\hspace{0.5em}\vec{w}\hspace{0.5em}
\cdots\hspace{1em}\vec{v}\hspace{0.5em} \cdots])
\end{equation*}
</div>
</li>
<li id="li-608">
<p id="p-859">Adding a multiple of a column to another column:</p>
<div class="displaymath">
\begin{equation*}
\det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}
\cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])=
\det([\cdots\hspace{0.5em}\vec{v}+c\vec{w}\hspace{0.5em}
\cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])
\end{equation*}
</div>
</li>
</ol></article><article class="remark remark-like" id="remark-17"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">5.1.13</span><span class="period">.</span>
</h6>
<p id="p-860">The determinants of row operation matrices may be computed by manipulating columns to reduce each matrix to the identity:</p>
<ul class="disc">
<li id="li-609"><p id="p-861">Scaling a row: \(\left[\begin{array}{cccc}
1 & 0 & 0 &0 \\
0 & c & 0 &0\\
0 & 0 & 1 &0 \\
0 & 0 & 0 & 0
\end{array}\right]\)</p></li>
<li id="li-610"><p id="p-862">Swapping rows: \(\left[\begin{array}{cccc}
0 & 1 & 0 &0 \\
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]\)</p></li>
<li id="li-611"><p id="p-863">Adding a row multiple to another row: \(\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & c & 0\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]\)</p></li>
</ul></article><article class="fact theorem-like" id="fact-26"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">5.1.14</span><span class="period">.</span>
</h6>
<p id="p-864">Thus we can also use row operations to simplify determinants:</p>
<ol class="decimal">
<li id="li-612"><p id="p-865">Multiplying rows by scalars: \(\det\left[\begin{array}{c}\vdots\\cR\\\vdots\end{array}\right]=
c\det\left[\begin{array}{c}\vdots\\R\\\vdots\end{array}\right]\)</p></li>
<li id="li-613"><p id="p-866">Swapping two rows: \(\det\left[\begin{array}{c}\vdots\\R\\\vdots\\S\\\vdots\end{array}\right]=
-\det\left[\begin{array}{c}\vdots\\S\\\vdots\\R\\\vdots\end{array}\right]\)</p></li>
<li id="li-614"><p id="p-867">Adding multiples of rows to other rows: \(\det\left[\begin{array}{c}\vdots\\R\\\vdots\\S\\\vdots\end{array}\right]=
\det\left[\begin{array}{c}\vdots\\R+cS\\\vdots\\S\\\vdots\end{array}\right]\)</p></li>
</ol></article><article class="observation remark-like" id="observation-30"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">5.1.15</span><span class="period">.</span>
</h6>
<p id="p-868">So we may compute the determinant of \(\left[\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}\right]\) by manipulating its rows/columns to reduce the matrix to \(I\text{:}\)</p>
<div class="displaymath">
\begin{align*}
\det\left[\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}\right]
&=
2 \det \left[\begin{array}{cc} 1 & 2 \\ 2 & 3 \end{array}\right]\\
&=
%2 \det \left[\begin{array}{cc} 1 & 2 \\ 2-2(1) & 3-2(2)\end{array}\right]=
2 \det \left[\begin{array}{cc} 1 & 2 \\ 0 & -1 \end{array}\right]\\
&=
%2(-1) \det \left[\begin{array}{cc} 1 & -2 \\ 0 & +1 \end{array}\right]=
-2 \det \left[\begin{array}{cc} 1 & -2 \\ 0 & 1 \end{array}\right]\\
&=
%-2 \det \left[\begin{array}{cc} 1+2(0) & -2+2(1) \\ 0 & 1\end{array}\right] =
-2 \det \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]\\
&=
%-2\det I =
%-2(1) =
-2
\end{align*}
</div></article><section class="exercises" id="exercises-21"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">5.1.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-101"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-869">Let \(A\) be a \(4 \times 4\) matrix with determinant \(-2 \text{.}\)</p>
<ol class="lower-alpha">
<li id="li-615">Let \(M\) be the matrix obtained from \(A\) by applying the row operation \(R_1 \to -4R_1 \text{.}\) What is \(\operatorname{det}\ M\text{?}\)</li>
<li id="li-616">Let \(N\) be the matrix obtained from \(A\) by applying the row operation \(R_1 \to R_1 + -3R_2 \text{.}\) What is \(\operatorname{det}\ N\text{?}\)</li>
<li id="li-617">Let \(P\) be the matrix obtained from \(A\) by applying the row operation \(R_1 \leftrightarrow R_4 \text{.}\) What is \(\operatorname{det}\ P\text{?}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-101" id="answer-101"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-101"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-618">\(\displaystyle \operatorname{det}\ M= 8 \)</li>
<li id="li-619">\(\displaystyle \operatorname{det}\ N= -2 \)</li>
<li id="li-620">\(\displaystyle \operatorname{det}\ P= 2 \)</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-102"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-870">Let \(A\) be a \(4 \times 4\) matrix with determinant \(-4 \text{.}\)</p>
<ol class="lower-alpha">
<li id="li-621">Let \(N\) be the matrix obtained from \(A\) by applying the row operation \(R_4 \to -5R_4 \text{.}\) What is \(\operatorname{det}\ N\text{?}\)</li>
<li id="li-622">Let \(P\) be the matrix obtained from \(A\) by applying the row operation \(R_3 \to R_3 + -3R_2 \text{.}\) What is \(\operatorname{det}\ P\text{?}\)</li>
<li id="li-623">Let \(C\) be the matrix obtained from \(A\) by applying the row operation \(R_4 \leftrightarrow R_3 \text{.}\) What is \(\operatorname{det}\ C\text{?}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-102" id="answer-102"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-102"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-624">\(\displaystyle \operatorname{det}\ N= 20 \)</li>
<li id="li-625">\(\displaystyle \operatorname{det}\ P= -4 \)</li>
<li id="li-626">\(\displaystyle \operatorname{det}\ C= 4 \)</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-103"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-871">Let \(A\) be a \(4 \times 4\) matrix with determinant \(7 \text{.}\)</p>
<ol class="lower-alpha">
<li id="li-627">Let \(B\) be the matrix obtained from \(A\) by applying the row operation \(R_3 \leftrightarrow R_1 \text{.}\) What is \(\operatorname{det}\ B\text{?}\)</li>
<li id="li-628">Let \(P\) be the matrix obtained from \(A\) by applying the row operation \(R_2 \to R_2 + 5R_3 \text{.}\) What is \(\operatorname{det}\ P\text{?}\)</li>
<li id="li-629">Let \(M\) be the matrix obtained from \(A\) by applying the row operation \(R_1 \to 3R_1 \text{.}\) What is \(\operatorname{det}\ M\text{?}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-103" id="answer-103"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-103"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-630">\(\displaystyle \operatorname{det}\ B= -7 \)</li>
<li id="li-631">\(\displaystyle \operatorname{det}\ P= 7 \)</li>
<li id="li-632">\(\displaystyle \operatorname{det}\ M= 21 \)</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-104"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-872">Let \(A\) be a \(4 \times 4\) matrix with determinant \(5 \text{.}\)</p>
<ol class="lower-alpha">
<li id="li-633">Let \(N\) be the matrix obtained from \(A\) by applying the row operation \(R_1 \to R_1 + -3R_3 \text{.}\) What is \(\operatorname{det}\ N\text{?}\)</li>
<li id="li-634">Let \(B\) be the matrix obtained from \(A\) by applying the row operation \(R_2 \leftrightarrow R_3 \text{.}\) What is \(\operatorname{det}\ B\text{?}\)</li>
<li id="li-635">Let \(M\) be the matrix obtained from \(A\) by applying the row operation \(R_4 \to -4R_4 \text{.}\) What is \(\operatorname{det}\ M\text{?}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-104" id="answer-104"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-104"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-636">\(\displaystyle \operatorname{det}\ N= 5 \)</li>
<li id="li-637">\(\displaystyle \operatorname{det}\ B= -5 \)</li>
<li id="li-638">\(\displaystyle \operatorname{det}\ M= -20 \)</li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-105"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-873">Let \(A\) be a \(4 \times 4\) matrix with determinant \(6 \text{.}\)</p>
<ol class="lower-alpha">
<li id="li-639">Let \(N\) be the matrix obtained from \(A\) by applying the row operation \(R_3 \to -3R_3 \text{.}\) What is \(\operatorname{det}\ N\text{?}\)</li>
<li id="li-640">Let \(Q\) be the matrix obtained from \(A\) by applying the row operation \(R_1 \leftrightarrow R_2 \text{.}\) What is \(\operatorname{det}\ Q\text{?}\)</li>
<li id="li-641">Let \(C\) be the matrix obtained from \(A\) by applying the row operation \(R_1 \to R_1 + 5R_2 \text{.}\) What is \(\operatorname{det}\ C\text{?}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-105" id="answer-105"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-105"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-642">\(\displaystyle \operatorname{det}\ N= -18 \)</li>
<li id="li-643">\(\displaystyle \operatorname{det}\ Q= -6 \)</li>
<li id="li-644">\(\displaystyle \operatorname{det}\ C= 6 \)</li>
</ol></div></div>
</div></article><p id="p-874"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/G1/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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