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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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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="G2"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">5.2</span> <span class="title">Computing Determinants (G2)</span>
</h2>
<article class="remark remark-like" id="remark-18"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">5.2.1</span><span class="period">.</span>
</h6>
<p id="p-875">We've seen that row reducing all the way into RREF gives us a method of computing determinants.</p>
<p id="p-876">However, we learned in module E that this can be tedious for large matrices. Thus, we will try to figure out how to turn the determinant of a larger matrix into the determinant of a smaller matrix.</p></article><article class="activity project-like" id="activity-130"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.2.1</span><span class="period">.</span>
</h6>
<p id="p-877">The following image illustrates the transformation of the unit cube by the matrix \(\left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 0 & 1\end{array}\right]\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}
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\draw[thin,gray,->] (0,0,0) -- (3,0,0);
\draw[thin,gray,->] (0,0,0) -- (0,2,0);
\draw[thin,gray,->] (0,0,0) -- (0,0,2);
%(y,z,x)
\draw[blue] (1,0,1) -- (4,0,2) -- (3,0,1);
\draw[blue] (1,1,0) -- (2,1,1) -- (5,1,2) -- (4,1,1) -- (1,1,0);
\draw[blue] (1,0,1) -- +(1,1,0);
\draw[blue] (4,0,2) -- +(1,1,0);
\draw[blue] (3,0,1) -- +(1,1,0);
\draw[purple,thick,->] (0,0,0) -- (1,1,0)
node[above left]{\tiny\(\left[\begin{array}{c} 0 \\ 1 \\ 1\end{array}\right]\)};
\draw[purple,thick,->] (0,0,0) -- (1,0,1)
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\draw[purple,thick,->] (0,0,0) -- (3,0,1)
node[above left]{\tiny\(\left[\begin{array}{c} 1 \\ 3 \\ 0\end{array}\right]\)};
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</pre>
<p id="p-878">Recall that for this solid \(V=Bh\text{,}\) where \(h\) is the height of the solid and \(B\) is the area of its parallelogram base. So what must its volume be?</p>
<ol class="lower-alpha">
<li id="li-645"><p id="p-879">\(\displaystyle \det \left[\begin{array}{cc} 1 & 1 \\ 1 & 3 \end{array}\right]\)</p></li>
<li id="li-646"><p id="p-880">\(\displaystyle \det \left[\begin{array}{cc} 1 & 0 \\ 3 & 1 \end{array}\right]\)</p></li>
<li id="li-647"><p id="p-881">\(\displaystyle \det \left[\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right]\)</p></li>
<li id="li-648"><p id="p-882">\(\displaystyle \det \left[\begin{array}{cc} 1 & 3 \\ 0 & 0 \end{array}\right]\)</p></li>
</ol></article><article class="fact theorem-like" id="fact-27"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">5.2.2</span><span class="period">.</span>
</h6>
<p id="p-883">If row \(i\) contains all zeros except for a \(1\) on the main (upper-left to lower-right) diagonal, then both column and row \(i\) may be removed without changing the value of the determinant.</p>
<div class="displaymath">
\begin{equation*}
\det \left[\begin{array}{cccc}
3 & {\color{red} 2} & -1 & 3 \\
{\color{red} 0} & {\color{red} 1}
& {\color{red} 0} & {\color{red} 0} \\
-1 & {\color{red} 4} & 1 & 0 \\
5 & {\color{red} 0} & 11 & 1
\end{array}\right] =
\det \left[\begin{array}{ccc}
3 & -1 & 3 \\
-1 & 1 & 0 \\
5 & 11 & 1
\end{array}\right]
\end{equation*}
</div>
<p id="p-884">Since row and column operations affect the determinants in the same way, the same technique works for a column of all zeros except for a \(1\) on the main diagonal.</p>
<div class="displaymath">
\begin{equation*}
\det \left[\begin{array}{cccc}
3 & {\color{red} 0} & -1 & 5 \\
{\color{red} 2} & {\color{red} 1} & {\color{red} 4} &
{\color{red} 0} \\
-1 & {\color{red} 0} & 1 & 11 \\
3 & {\color{red} 0} & 0 & 1
\end{array}\right] =
\det \left[\begin{array}{ccc}
3 & -1 & 5 \\
-1 & 1 & 11 \\
3 & 0 & 1
\end{array}\right]
\end{equation*}
</div></article><article class="activity project-like" id="activity-131"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.2.2</span><span class="period">.</span>
</h6>
<p id="p-885">Remove an appropriate row and column of \(\det \left[\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 5 & 12 \\ 3 & 2 & -1 \end{array}\right]\) to simplify the determinant to a \(2\times 2\) determinant.</p></article><article class="activity project-like" id="activity-132"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.2.3</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-39"><p id="p-886">Simplify \(\det \left[\begin{array}{ccc} 0 & 3 & -2 \\ 2 & 5 & 12 \\ 0 & 2 & -1 \end{array}\right]\) to a multiple of a \(2\times 2\) determinant by first doing the following:</p></div>
<article class="task exercise-like" id="task-94"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-887">Factor out a \(2\) from a column.</p></article><article class="task exercise-like" id="task-95"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-888">Swap rows or columns to put a \(1\) on the main diagonal.</p></article></article><article class="activity project-like" id="activity-133"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.2.4</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-40"><p id="p-889">Simplify \(\det \left[\begin{array}{ccc} 4 & -2 & 2 \\ 3 & 1 & 4 \\ 1 & -1 & 3\end{array}\right]\) to a multiple of a \(2\times 2\) determinant by first doing the following:</p></div>
<article class="task exercise-like" id="task-96"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-890">Use row/column operations to create two zeroes in the same row or column.</p></article><article class="task exercise-like" id="task-97"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-891">Factor/swap as needed to get a row/column of all zeroes except a \(1\) on the main diagonal.</p></article></article><article class="observation remark-like" id="observation-31"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">5.2.3</span><span class="period">.</span>
</h6>
<p id="p-892">Using row/column operations, you can introduce zeros and reduce dimension to whittle down the determinant of a large matrix to a determinant of a smaller matrix.</p>
<div class="displaymath">
\begin{align*}
\det\left[\begin{array}{cccc}
4 & 3 & 0 & 1 \\
2 & -2 & 4 & 0 \\
-1 & 4 & 1 & 5 \\
2 & 8 & 0 & 3
\end{array}\right]
&=
\det\left[\begin{array}{cccc}
4 & 3 & {\color{red} 0} & 1 \\
6 & -18 & {\color{red} 0} & -20 \\
{\color{red} -1} & {\color{red} 4} &
{\color{red} 1} & {\color{red} 5} \\
2 & 8 & {\color{red} 0} & 3
\end{array}\right]
=
\det\left[\begin{array}{ccc}
4 & 3 & 1 \\
6 & -18 & -20 \\
2 & 8 & 3
\end{array}\right]\\
&=\dots=
-2\det\left[\begin{array}{ccc}
{\color{red} 1} & {\color{red} 3} & {\color{red} 4} \\
{\color{red} 0} & 21 & 43 \\
{\color{red} 0} & -1 & -10
\end{array}\right]
=
-2\det\left[\begin{array}{cc} 21 & 43 \\ -1 & -10 \end{array}\right]\\
&= \dots=
-2\det\left[\begin{array}{cc}
-167 & {\color{red}{21}} \\
{\color{red} 0} & {\color{red} 1}
\end{array}\right]
= -2\det[-167]\\
&=-2(-167)\det(I)=
334
\end{align*}
</div></article><article class="activity project-like" id="activity-134"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.2.5</span><span class="period">.</span>
</h6>
<p id="p-893">Rewrite</p>
<div class="displaymath">
\begin{equation*}
\det \left[\begin{array}{cccc} 2 & 1 & -2 & 1 \\ 3 & 0 & 1 & 4
\\ -2 & 2 & 3 & 0 \\ -2 & 0 & -3 & -3 \end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">as a multiple of a determinant of a \(3\times3\) matrix.</p></article><article class="activity project-like" id="activity-135"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.2.6</span><span class="period">.</span>
</h6>
<p id="p-894">Compute \(\det\left[\begin{array}{cccc}
2 & 3 & 5 & 0 \\
0 & 3 & 2 & 0 \\
1 & 2 & 0 & 3 \\
-1 & -1 & 2 & 2
\end{array}\right]\) by using any combination of row/column operations.</p></article><article class="observation remark-like" id="observation-32"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">5.2.4</span><span class="period">.</span>
</h6>
<p id="p-895">Another option is to take advantage of the fact that the determinant is linear in each row or column. This approach is called <dfn class="terminology">Laplace expansion</dfn> or <dfn class="terminology">cofactor expansion</dfn>.</p>
<p id="p-896">For example, since \(\color{blue}{
\left[\begin{array}{ccc} 1 & 2 & 4 \end{array}\right]
=
1\left[\begin{array}{ccc} 1 & 0 & 0 \end{array}\right]
+
2\left[\begin{array}{ccc} 0 & 1 & 0 \end{array}\right]
+
4\left[\begin{array}{ccc} 0 & 0 & 1 \end{array}\right]}
\text{,}\)</p>
<div class="displaymath">
\begin{align*}
\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ {\color{blue} 1} & {\color{blue} 2} & {\color{blue} 4} \end{array}\right] &=
{\color{blue} 1}\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ {\color{blue} 1} & {\color{blue} 0} & {\color{blue} 0} \end{array}\right] +
{\color{blue} 2}\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ {\color{blue} 0} & {\color{blue} 1} & {\color{blue} 0} \end{array}\right] +
{\color{blue} 4}\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ {\color{blue} 0} & {\color{blue} 0} & {\color{blue} 1} \end{array}\right]\\
&= -1\det \left[\begin{array}{ccc} 5 & 3 & 2 \\ 5 & 3 & -1 \\ 0 & 0 & 1 \end{array}\right]
-2\det \left[\begin{array}{ccc} 2 & 5 & 3 \\ -1 & 5 & 3 \\ 0 & 0 & 1 \end{array}\right] +
4\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ 0 & 0 & 1 \end{array}\right]\\
&= -\det \left[\begin{array}{cc} 5 & 3 \\ 5 & 3 \end{array}\right]
-2 \det \left[\begin{array}{cc} 2 & 5 \\ -1 & 5 \end{array}\right]
+4 \det \left[\begin{array}{cc} 2 & 3 \\ -1 & 3 \end{array}\right]
\end{align*}
</div></article><article class="observation remark-like" id="observation-33"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">5.2.5</span><span class="period">.</span>
</h6>
<p id="p-897">Applying Laplace expansion to a \(2 \times 2\) matrix yields a short formula you may have seen:</p>
<div class="displaymath">
\begin{equation*}
\det \left[\begin{array}{cc} {\color{blue} a} & {\color{blue} b} \\ c & d \end{array}\right]
=
{\color{blue} a}\det \left[\begin{array}{cc} {\color{blue} 1} & {\color{blue} 0} \\
c & d \end{array}\right]
+
{\color{blue} b} \det \left[\begin{array}{cc} {\color{blue} 0} & {\color{blue} 1} \\
c & d \end{array}\right]
=
a\det \left[\begin{array}{cc} {\color{red} 1} & {\color{red} 0} \\
{\color{red} c} & d \end{array}\right]
-
b \det \left[\begin{array}{cc} {\color{red} 1} & {\color{red} 0} \\
{\color{red} d} & c \end{array}\right]
=
ad-bc\text{.}
\end{equation*}
</div>
<p id="p-898">There are formulas for the determinants of larger matrices, but they can be pretty tedious to use. For example, writing out a formula for a \(4\times 4\) determinant would require 24 different terms!</p>
<div class="displaymath">
\begin{equation*}
\det\left[\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{array}\right]
=
a_{11}(a_{22}(a_{33}a_{44}-a_{43}a_{34})-a_{23}(a_{32}a_{44}-a_{42}a_{34})+\dots)+\dots
\end{equation*}
</div>
<p id="p-899">So this is why we either use Laplace expansion or row/column operations directly.</p></article><article class="activity project-like" id="activity-136"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.2.7</span><span class="period">.</span>
</h6>
<p id="p-900">Based on the previous activities, which technique is easier for computing determinants?</p>
<ol class="lower-alpha">
<li id="li-649"><p id="p-901">Memorizing formulas.</p></li>
<li id="li-650"><p id="p-902">Using row/column operations.</p></li>
<li id="li-651"><p id="p-903">Laplace expansion.</p></li>
<li id="li-652"><p id="p-904">Some other technique (be prepared to describe it).</p></li>
</ol></article><article class="activity project-like" id="activity-137"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.2.8</span><span class="period">.</span>
</h6>
<p id="p-905">Use your preferred technique to compute \(\det\left[\begin{array}{cccc}
4 & -3 & 0 & 0 \\
1 & -3 & 2 & -1 \\
3 & 2 & 0 & 3 \\
0 & -3 & 2 & -2
\end{array}\right]
\text{.}\)</p></article><section class="exercises" id="exercises-22"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">5.2.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-106"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-906">Show how to compute the determinant of the matrix</p>
<div class="displaymath">
\begin{equation*}
A= \left[\begin{array}{cccc}
-5 & -1 & 5 & 0 \\
-2 & 5 & -3 & 0 \\
-1 & 0 & 3 & 3 \\
-4 & -6 & -6 & -1
\end{array}\right] .\text{.}
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-106" id="answer-106"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-106"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
\operatorname{det}\ \left[\begin{array}{cccc}
-5 & -1 & 5 & 0 \\
-2 & 5 & -3 & 0 \\
-1 & 0 & 3 & 3 \\
-4 & -6 & -6 & -1
\end{array}\right] = -1141
\end{equation*}
</div></div></div>
</div></article><article class="exercise exercise-like" id="exercise-107"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-907">Show how to compute the determinant of the matrix</p>
<div class="displaymath">
\begin{equation*}
A= \left[\begin{array}{cccc}
5 & 0 & 1 & 1 \\
-6 & -5 & -3 & 0 \\
2 & 3 & 6 & -2 \\
-6 & 1 & -1 & 0
\end{array}\right] .\text{.}
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-107" id="answer-107"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-107"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
\operatorname{det}\ \left[\begin{array}{cccc}
5 & 0 & 1 & 1 \\
-6 & -5 & -3 & 0 \\
2 & 3 & 6 & -2 \\
-6 & 1 & -1 & 0
\end{array}\right] = -156
\end{equation*}
</div></div></div>
</div></article><article class="exercise exercise-like" id="exercise-108"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-908">Show how to compute the determinant of the matrix</p>
<div class="displaymath">
\begin{equation*}
A= \left[\begin{array}{cccc}
1 & -4 & 2 & 3 \\
-6 & 4 & -5 & -3 \\
-1 & -1 & 1 & 4 \\
-1 & 0 & 0 & 2
\end{array}\right] .\text{.}
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-108" id="answer-108"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-108"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
\operatorname{det}\ \left[\begin{array}{cccc}
1 & -4 & 2 & 3 \\
-6 & 4 & -5 & -3 \\
-1 & -1 & 1 & 4 \\
-1 & 0 & 0 & 2
\end{array}\right] = -11
\end{equation*}
</div></div></div>
</div></article><article class="exercise exercise-like" id="exercise-109"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-909">Show how to compute the determinant of the matrix</p>
<div class="displaymath">
\begin{equation*}
A= \left[\begin{array}{cccc}
4 & 3 & -2 & -5 \\
2 & -1 & 6 & 0 \\
0 & 0 & 1 & -3 \\
3 & 5 & 5 & -2
\end{array}\right] .\text{.}
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-109" id="answer-109"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-109"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
\operatorname{det}\ \left[\begin{array}{cccc}
4 & 3 & -2 & -5 \\
2 & -1 & 6 & 0 \\
0 & 0 & 1 & -3 \\
3 & 5 & 5 & -2
\end{array}\right] = -471
\end{equation*}
</div></div></div>
</div></article><article class="exercise exercise-like" id="exercise-110"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-910">Show how to compute the determinant of the matrix</p>
<div class="displaymath">
\begin{equation*}
A= \left[\begin{array}{cccc}
3 & -5 & -6 & 0 \\
-2 & -2 & -4 & 0 \\
-2 & -6 & 4 & 0 \\
2 & -3 & 3 & 2
\end{array}\right] .\text{.}
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-110" id="answer-110"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-110"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
\operatorname{det}\ \left[\begin{array}{cccc}
3 & -5 & -6 & 0 \\
-2 & -2 & -4 & 0 \\
-2 & -6 & 4 & 0 \\
2 & -3 & 3 & 2
\end{array}\right] = -448
\end{equation*}
</div></div></div>
</div></article><p id="p-911"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/G2/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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