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week13.html
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<body><div class="sidenav normalwidth"><button style="border:none; background-color: Transparent;" onclick="showhidemenu()" title="Toggle toc"><span style="font-size: 30px;">☰</span></button><button class="openbs" title="Open buttons">+</button><button class="closebs" title="Close buttons">-</button><ul class="leftmenu" style="display: none;"><li><a class="kap" href="introduction.html#159ca076-dc4e-4bb9-83f5-a5ddb8ff2299"><b>1</b> Introduction</a></li><a href="#kap1" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap1" class="collapse"><li><a href="introduction.html#ae495219-240d-4cf1-9a9b-29918804ef56">1.1 Readings</a></li><li><a href="introduction.html#26671802-fb3b-4273-8e82-12d04b3b35a4">1.2 Notes</a></li><li><a href="introduction.html#3fc447d4-9ac7-439e-afc9-6c5ad35b2e24">1.3 Exercises</a></li></ul><li><a class="kap" href="week1.html#3a6133ac-4a7b-47d1-9ea5-47ca81f168f2"><b>2</b> Course week 1</a></li><a href="#kap2" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap2" class="collapse"><li><a href="week1.html#cbb4e3bf-894c-49ad-90c5-58d0ed1e7cc3">2.1 Readings</a></li><li><a href="week1.html#d018418e-a17c-4b70-b799-f34e0298333b">2.2 Notes</a></li><li><a href="week1.html#e010e5dd-5121-4b83-8e89-d32234dee55e">2.3 Exercises</a></li></ul><li><a class="kap" href="week2.html#5c85fc37-321e-4707-9e48-bdb3f9e4a5bd"><b>3</b> Course week 2</a></li><a href="#kap3" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap3" class="collapse"><li><a href="week2.html#8ec6af3d-7c28-4353-89f8-1992db17e7e8">3.1 Readings</a></li><li><a href="week2.html#b18e048c-25be-4300-9ed0-4407e46d540c">3.2 Notes</a></li><li><a href="week2.html#cca75863-ec45-459f-a0f4-3a965a60422e">3.3 Exercises</a></li><li><a href="week2.html#ee2537cc-717b-4463-8904-87085c60123a">3.4 Feedback</a></li></ul><li><a class="kap" href="week3.html#9beeae3e-d65c-4020-b00a-c526ceadf899"><b>4</b> Course week 3</a></li><a href="#kap4" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap4" class="collapse"><li><a href="week3.html#6bef6f20-56d9-4f57-803f-0ba61b2540ac">4.1 Readings</a></li><li><a href="week3.html#7a92877c-98c7-4bdd-9e2f-762cb37563e8">4.2 Notes</a></li><li><a href="week3.html#b4fcf09c-686b-41fc-8a20-854b9c21bf31">4.3 Exercises</a></li><li><a href="week3.html#fe6dc55b-aaac-49aa-9a44-55212b41273d">4.4 Feedback</a></li></ul><li><a class="kap" href="week4.html#2402c1cb-8568-4fb6-bdbe-b6485e20d1ca"><b>5</b> Course week 4</a></li><a href="#kap5" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap5" class="collapse"><li><a href="week4.html#1e2ed3d8-2100-4005-899d-ebde86500961">5.1 Readings</a></li><li><a href="week4.html#34298628-e47f-4a13-a2e8-ee6d069fc8b5">5.2 Notes</a></li><li><a href="week4.html#23a8a6e0-68cc-464c-a101-71cdfbbe4353">5.3 Problems</a></li><li><a href="week4.html#497d6801-a7db-445d-b745-3c9c166483b5">5.4 Feedback</a></li></ul><li><a class="kap" href="week5.html#881f0ad5-1f5b-406f-b1f3-cf97397cdfe3"><b>6</b> Course week 5</a></li><a href="#kap6" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap6" class="collapse"><li><a href="week5.html#f420a408-715e-462b-b2c1-59a979e8b14e">6.1 Readings</a></li><li><a href="week5.html#718a53d7-d8cf-4f5e-8ecf-6561f6f394b8">6.2 Notes</a></li><li><a href="week5.html#08659956-822b-4a08-9e5f-f727bd863256">6.3 Problems</a></li><li><a href="week5.html#74ef3f61-c1c2-42f7-b3c5-dbf69578da08">6.4 Feedback</a></li></ul><li><a class="kap" href="week6.html#b028fe91-09aa-484e-af70-e887bdccbf90"><b>7</b> Course week 6</a></li><a href="#kap7" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap7" class="collapse"><li><a href="week6.html#5f5a6489-a309-4ba4-9cc2-e37611e8e350">7.1 Readings</a></li><li><a href="week6.html#bebc3711-369c-44c2-a9a1-23c8bc9900b0">7.2 Notes</a></li><li><a href="week6.html#2619640f-84f5-4826-8212-f383e4451cb3">7.3 Problems</a></li><li><a href="week6.html#8626687f-527d-42cc-94eb-5f19d20ba029">7.4 Feedback</a></li></ul><li><a class="kap" href="week7.html#c4dc25ad-24a4-45e7-b4c5-e59016fd3dcc"><b>8</b> Course week 7</a></li><a href="#kap8" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap8" class="collapse"><li><a href="week7.html#ee3b5046-f23c-41e8-beb7-08e80a78ebca">8.1 Readings</a></li><li><a href="week7.html#98cafeeb-e279-4a89-a571-d5fc17e6824b">8.2 Notes</a></li><li><a href="week7.html#fdd931f4-395e-40ca-81b8-c3186c52d21d">8.3 Problems</a></li><li><a href="week7.html#4de6c567-85d9-4936-92e4-4f323d10a16b">8.4 Feedback</a></li></ul><li><a class="kap" href="week8.html#a6129057-0dca-47e2-a908-acd3fc725bca"><b>9</b> Course week 8</a></li><a href="#kap9" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap9" class="collapse"><li><a href="week8.html#91a7f924-2cc3-46e0-99d5-84b72cd25982">9.1 Readings</a></li><li><a href="week8.html#2227f711-3610-4e0e-b835-539a1c67559b">9.2 Notes</a></li><li><a href="week8.html#a580698f-3a0d-489f-a786-09490c52da6a">9.3 Problems</a></li><li><a href="week8.html#f636cddc-14ea-46dc-921a-fc8de482f125">9.4 Feedback</a></li></ul><li><a class="kap" href="week9.html#1673477a-155b-4ee5-8c3c-524c4da8f0da"><b>10</b> Course week 9</a></li><a href="#kap10" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap10" class="collapse"><li><a href="week9.html#8d4c9c5d-c542-4cd9-8401-e1a52df4286f">10.1 Readings</a></li><li><a href="week9.html#4dec455a-332a-4ae1-99a8-9c55b704483d">10.2 Notes</a></li><li><a href="week9.html#af8ac1d5-5cc3-4c7d-8eeb-bd6086a2beb9">10.3 Problems</a></li><li><a href="week9.html#9fb03b93-76bb-4df9-ba3b-3076a33a3c84">10.4 Feedback</a></li></ul><li><a class="kap" href="week10.html#9a44ce9a-039b-43ea-bab2-a8ca6d6ad271"><b>11</b> Course week 10</a></li><a href="#kap11" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap11" class="collapse"><li><a href="week10.html#0a130e84-47b6-42a1-b4a0-c045425900f4">11.1 Readings</a></li><li><a href="week10.html#e0c9a4d0-1c3b-4411-9a42-5eac771e0265">11.2 Notes</a></li><li><a href="week10.html#13ab74d2-e187-4b74-a77c-c9599ae7c0ed">11.3 Problems</a></li><li><a href="week10.html#7b803349-1c7b-40d6-9855-33a3e7b15c17">11.4 Feedback</a></li></ul><li><a class="kap" href="week11.html#bad3f0c8-0554-443d-bd04-1392582ea095"><b>12</b> Course week 11</a></li><a href="#kap12" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap12" class="collapse"><li><a href="week11.html#8a687e45-dd04-4a19-a682-b8573340d40e">12.1 Readings</a></li><li><a href="week11.html#03a71c8c-2af2-4746-bbb6-8efe3d3e5e6b">12.2 Notes</a></li><li><a href="week11.html#da7ad12d-8b44-4cfe-a32f-9f8954d6f1bb">12.3 Problems</a></li><li><a href="week11.html#f729eb4e-feb9-4840-a28e-4524cefeda12">12.4 Feedback</a></li></ul><li><a class="kap" href="week12.html#4edeeb01-8694-4c0d-bab3-cea029123630"><b>13</b> Course week 12</a></li><a href="#kap13" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap13" class="collapse"><li><a href="week12.html#3a9c26cd-489b-4a83-84e0-b7ce17925a99">13.1 Readings</a></li><li><a href="week12.html#ae4d4c1f-1399-464e-af22-d908c76bb143">13.2 Notes</a></li><li><a href="week12.html#caf55b0e-7ea3-4ccd-84fb-0c2dfaa25472">13.3 Problems</a></li><li><a href="week12.html#ac0eb5fa-eb33-436e-95b9-d3bf33460ad2">13.4 Feedback</a></li></ul><li><a class="kap" href="week13.html#f0e1b4c2-d02a-4740-bc27-8b653fedf832"><b>14</b> Course week 13</a></li><a href="#kap14" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap14" class="collapse"><li><a href="week13.html#36ebe4f7-f0c7-46e4-8404-b0bb54f694c9">14.1 Readings</a></li><li><a href="week13.html#b6ef6597-50c4-47d2-a2d0-15334207b7d0">14.2 Notes</a></li><li><a href="week13.html#7b891666-2d5c-4f9a-80f9-adfff76ce1f4">14.3 Problems</a></li><li><a href="week13.html#988cdc9c-6611-42fc-b4eb-370b70fb44c1">14.4 Feedback</a></li></ul><li><a class="kap" href="week14.html#96e609e7-8f1b-4f46-8653-af953c4e9823"><b>15</b> Course week 14</a></li><a href="#kap15" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap15" class="collapse"><li><a href="week14.html#90ef85b9-1287-4e2c-9a13-5b8f4bca1584">15.1 Readings</a></li><li><a href="week14.html#3350aecb-60a7-4229-97a1-c063c9336eeb">15.2 Notes</a></li><li><a href="week14.html#aeadc6cf-2833-4923-b0f5-14a9ab76ce1e">15.3 Problems</a></li></ul></ul></div><div class="main normalmargin"><div style="margin-top:20px"></div><div style="margin-top:20px"></div><h1 id="f0e1b4c2-d02a-4740-bc27-8b653fedf832">14<span style="float:right;">Course week 13</span></h1><div style="margin-top:20px"></div><span id="sec14.1"></span><h2 id="36ebe4f7-f0c7-46e4-8404-b0bb54f694c9">14.1 Readings</h2>
Read sections 12.6–12.7 and 13.1 in the textbook.<div style="margin-top:20px"></div><span id="sec14.2"></span><h2 id="b6ef6597-50c4-47d2-a2d0-15334207b7d0">14.2 Notes</h2>
<b>Rules for matrix multiplication</b>
<div class="frameit">
<ol class="number"><li id="ite14.1"> <span class="math"></span><script type="math/tex">(\mathbf{A}\mathbf{B})\mathbf{C} = \mathbf{A}(\mathbf{B}\mathbf{C}) </script>
</li><li id="ite14.2"> <span class="math"></span><script type="math/tex">\mathbf{A}(\mathbf{B}+\mathbf{C}) = \mathbf{AB} + \mathbf{AC} </script>
</li><li id="ite14.3"> <span class="math"></span><script type="math/tex">\mathbf{(A+B)C = AC+BC}</script>
</li><li id="ite14.4"> <span class="math"></span><script type="math/tex">(\alpha\mathbf{A})\mathbf{B} = \mathbf{A}(\alpha\mathbf{B}) = \alpha\mathbf{AB}</script>
</li></ol>
</div><div style="margin-top:20px"></div><b>The transpose</b>
<div class="frameit">
Transposition is an operation that swaps rows and columns of a matix. The result of transposing
<span class="math"></span><script type="math/tex">\mathbf{A}</script> is called the transpose of <span class="math"></span><script type="math/tex">\mathbf{A}</script>, and is written <span class="math"></span><script type="math/tex">\mathbf{A}'</script> (or <span class="math"></span><script type="math/tex">\mathbf{A}^\text{T}</script>).<div style="margin-top:20px"></div><div class="math"></div><script type="math/tex; mode=display">
\mathbf{A} =
\begin{pmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{pmatrix}
\qquad
\mathbf{A'} =
\begin{pmatrix}
a_{11} & a_{21} & \cdots & a_{m1} \\
a_{12} & a_{22} & \cdots & a_{m2} \\
\vdots & \vdots & & \vdots \\
a_{1n} & a_{2n} & \cdots & a_{mn}
\end{pmatrix}
</script><div style="margin-top:20px"></div>Note: For non-quadratic matrices <span class="math"></span><script type="math/tex">\mathbf{A}</script> and <span class="math"></span><script type="math/tex">\mathbf{A}'</script> have different dimensions.
</div><div style="margin-top:20px"></div><b>The determinant</b>
<div class="frameit">
The denominator in the solution for <span class="math"></span><script type="math/tex">x_1</script> and <span class="math"></span><script type="math/tex">x_2</script> is composed entirely of elements from <span class="math"></span><script type="math/tex">\mathbf{A}</script>.
It is called <span class="math"></span><script type="math/tex">\mathbf{A}</script>'s <em>determinant</em>, and must be <span class="math"></span><script type="math/tex">\neq 0.</script><div style="margin-top:20px"></div><div class="math"></div><script type="math/tex; mode=display">
\det\mathbf{A} = \vert\mathbf{A}\vert =
\begin{vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{vmatrix}
=a_{11}a_{22}-a_{21}a_{12}
</script><div style="margin-top:20px"></div>If <span class="math"></span><script type="math/tex">\vert\mathbf{A}\vert \neq 0</script> the system has exactly one solution. If <span class="math"></span><script type="math/tex">\vert\mathbf{A}\vert = 0</script> there are either 0 or <span class="math"></span><script type="math/tex">\infty</script> many solutions.
</div><div style="margin-top:20px"></div><b>Cramer's rule</b>
<div class="frameit">
The numerators in the solutions for <span class="math"></span><script type="math/tex">x_1</script> and <span class="math"></span><script type="math/tex">x_2</script> can be written as determinants:
<div class="math"></div><script type="math/tex; mode=display">
x_1 = \frac{ \begin{vmatrix}
b_{1} & a_{12} \\
b_{2} & a_{22}
\end{vmatrix} }{\vert\mathbf{A}\vert}
\qquad
x_2 = \frac{ \begin{vmatrix}
a_{11} & b_{1} \\
a_{21} & b_{2}
\end{vmatrix} }{\vert\mathbf{A}\vert}
</script><div style="margin-top:20px"></div>This is Cramer's rule for 2 equations with 2 unknowns (<span class="math"></span><script type="math/tex">n=2</script>).
</div><div style="margin-top:20px"></div><span id="sec14.3"></span><h2 id="7b891666-2d5c-4f9a-80f9-adfff76ce1f4">14.3 Problems</h2><div style="margin-top:20px"></div><span id="env14.1"></span><div class="exercise" data-count="14.1">
A department in an organisation has <span class="math"></span><script type="math/tex">N</script> employees, who vote to select a representative in the liaison committee. According the election rules, each employee has two votes (first and second priority), which they can use to vote for their colleagues in the department. The rules require that they vote for two different people with their two votes. The representative elected is the person with the most votes as the first priority, assuming that this person has at least three votes more than any other candidate. If this is not the case the second priority votes will be added to the first priority votes for all employees. The elected representative is then the person with the most votes, irrespective of how many votes the others have. If two or more people have the maximum number of votes, at this point, a new election will be held. The result of the vote is tallied using two matrices <span class="math"></span><script type="math/tex">\mathbf{A}</script> where an element <span class="math"></span><script type="math/tex">a_{ij} = 1</script> indicates that employee number <span class="math"></span><script type="math/tex">i</script> voted for employee number <span class="math"></span><script type="math/tex">j</script> as the first priority, while <span class="math"></span><script type="math/tex">a_{ij} = 0</script> means that <span class="math"></span><script type="math/tex">i</script> did not vote for <span class="math"></span><script type="math/tex">j</script>. Correspondingly, <span class="math"></span><script type="math/tex">\mathbf{B}</script> contains the second priority votes. It is assumed that all employees cast both of their votes in the election.
<ol class="number"><li id="ite14.5"> What is <span class="math"></span><script type="math/tex">\displaystyle\sum_{i=1}^{N} \sum_{j=1}^{N} a_{ij}</script> and <span class="math"></span><script type="math/tex">\displaystyle\sum_{i=1}^{N} \sum_{j=1}^{N} b_{ij}</script>?
</li><li id="ite14.6"> Are <span class="math"></span><script type="math/tex">\mathbf{A}</script> and <span class="math"></span><script type="math/tex">\mathbf{B}</script> necessarily symmetric? <a href="#9c273cf5-8e07-44b8-b86f-73f3fe50e4b0" class ="btn btn-default Hintbutton"data-toggle="collapse"></a> <div id=9c273cf5-8e07-44b8-b86f-73f3fe50e4b0 class = "collapse Hint envbuttons"> Think about what must be required of the elements if a matrix is symmetric. </div>
</li><li id="ite14.7"> Can <span class="math"></span><script type="math/tex">\mathbf{A}</script> and <span class="math"></span><script type="math/tex">\mathbf{B}</script> be symmetric? <a href="#d90b2a9c-b14f-464e-9345-6ba5c945480c" class ="btn btn-default Hintbutton"data-toggle="collapse"></a> <div id=d90b2a9c-b14f-464e-9345-6ba5c945480c class = "collapse Hint envbuttons"> Can you come up with a voting scenario where the matrix is symmetric? Does it matter whether <span class="math"></span><script type="math/tex">N</script> is even or odd? </div>
</li><li id="ite14.8"> What can you deduce about the elements on the diagonal (where <span class="math"></span><script type="math/tex">i = j</script>) in <span class="math"></span><script type="math/tex">\mathbf{A}</script> and <span class="math"></span><script type="math/tex">\mathbf{B}</script>?
</li><li id="ite14.9"> What is <span class="math"></span><script type="math/tex">\displaystyle\sum_{k=1}^{N} a_{kk}</script> and <span class="math"></span><script type="math/tex">\displaystyle\sum_{k=1}^{N} b_{kk}</script>?
</li><li id="ite14.10"> Express the condition for employee <span class="math"></span><script type="math/tex">k</script> winning the election in the first round in terms of the elements of <span class="math"></span><script type="math/tex">\mathbf{A}</script>.
</li><li id="ite14.11"> Express the condition for employee <span class="math"></span><script type="math/tex">k</script> winning in the second round in terms of the elements of <span class="math"></span><script type="math/tex">\mathbf{A}</script> and <span class="math"></span><script type="math/tex">\mathbf{B}</script>.
</li><li id="ite14.12"> Which of these matrices are valid candidates for <span class="math"></span><script type="math/tex">\mathbf{A}</script> and <span class="math"></span><script type="math/tex">\mathbf{B}</script>?
<a href="#79233241-6dc4-49fd-b5ec-e8c4d392ad1a" class ="btn btn-default" data-toggle="collapse">Show matrices</a><div id=79233241-6dc4-49fd-b5ec-e8c4d392ad1a class="collapse">
<div class="math"></div><script type="math/tex; mode=display">\mathbf{C} = \left(\begin{array}{rrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0
\end{array}\right) \qquad
\mathbf{D} = \left(\begin{array}{rrrrrrrr}
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right) \qquad
\mathbf{E} = \left(\begin{array}{rrrrrrrr}
0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0
\end{array}\right)</script>
</div>
</li><li id="ite14.13"> What is the outcome of the election if <span class="math"></span><script type="math/tex">\mathbf{A}</script> and <span class="math"></span><script type="math/tex">\mathbf{B}</script> are the following matrices?
<a href="#1f2fa97b-4557-405d-b6a0-a52bba58be40" class ="btn btn-default" data-toggle="collapse">Show matrices</a><div id=1f2fa97b-4557-405d-b6a0-a52bba58be40 class="collapse">
<div class="math"></div><script type="math/tex; mode=display">\mathbf{A} =
\left(\begin{array}{rrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right)
\qquad
\mathbf{B} = \left(\begin{array}{rrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right)</script>
</div>
</li></ol>
</div><div style="margin-top:20px"></div><span id="env14.2"></span><div class="exercise" data-count="14.2">
Two matrices <b>A</b> and <b>B</b> are given by:
<span class="math"></span><script type="math/tex">
\textbf{A} =
\begin{pmatrix*}[r]
3 & -5 & 7 \\
-3 & 1 & 2 \\
4 & -9 & 4
\end{pmatrix*}
</script> and
<span class="math"></span><script type="math/tex">
\textbf{B} =
\begin{pmatrix*}[r]
2 & 1 & 0 \\
3 & 2 & -2 \\
0 & 2 & -2
\end{pmatrix*}.
</script>
<div id="quiz1"> <div class="row quizquestion">
1. Calculate <b>AB</b>.<div style="margin-top:20px"></div> </div> <div class="row formatquizanswer"> <b>Your answer:</b> It is a <select class="formatquiz-type" onchange="formatQuizChangeType(1)"> <option value="none" selected></option> <option value="scalar">Scalar</option> <option value="columnvector">Column vector</option> <option value="rowvector">Row vector</option> <option value="matrix">Matrix</option> </select> <span class="formatquiz-scalar hidden"> which is </span> <span class="formatquiz-vector hidden"> with <select onchange="formatQuizChangeMatrixSize(1)"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> entries </span> <span class="formatquiz-matrix hidden"> with dimensions <select onchange="formatQuizChangeMatrixSize(1)" class="matrixrows"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> x <select onchange="formatQuizChangeMatrixSize(1)" class="matrixcolumns"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> </span> <span class="formatquiz-numbers hidden"> given by <table class="number-table"> </table> <input type="button" value="Check" onclick="formatQuizCheckAnswer(1, formatQuizProgram1)"> <span class="result"> <span class="number-errors"> </span> <span class="correct hidden"> <b>Correct!</b> </span> <span class="wrong hidden"> <b>Wrong.</b> </span> </span> <script type="text/javascript"> function formatQuizProgram1(matrix) {
if (matrixUID(matrix) != matrixUID(3, 3)) return false;
var result = true;
result = result && (equals(matrix[0][0], -9));
result = result && (equals(matrix[0][1], 7));
result = result && (equals(matrix[0][2], -4));
result = result && (equals(matrix[1][0], -3));
result = result && (equals(matrix[1][1], 3));
result = result && (equals(matrix[1][2], -6));
result = result && (equals(matrix[2][0], -19));
result = result && (equals(matrix[2][1], -6));
result = result && (equals(matrix[2][2], 10));
return result;
}
</script> </span> </div></div>
<div id="quiz2"> <div class="row quizquestion">
2. Calculate <b>BA</b>.<div style="margin-top:20px"></div> </div> <div class="row formatquizanswer"> <b>Your answer:</b> It is a <select class="formatquiz-type" onchange="formatQuizChangeType(2)"> <option value="none" selected></option> <option value="scalar">Scalar</option> <option value="columnvector">Column vector</option> <option value="rowvector">Row vector</option> <option value="matrix">Matrix</option> </select> <span class="formatquiz-scalar hidden"> which is </span> <span class="formatquiz-vector hidden"> with <select onchange="formatQuizChangeMatrixSize(2)"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> entries </span> <span class="formatquiz-matrix hidden"> with dimensions <select onchange="formatQuizChangeMatrixSize(2)" class="matrixrows"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> x <select onchange="formatQuizChangeMatrixSize(2)" class="matrixcolumns"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> </span> <span class="formatquiz-numbers hidden"> given by <table class="number-table"> </table> <input type="button" value="Check" onclick="formatQuizCheckAnswer(2, formatQuizProgram2)"> <span class="result"> <span class="number-errors"> </span> <span class="correct hidden"> <b>Correct!</b> </span> <span class="wrong hidden"> <b>Wrong.</b> </span> </span> <script type="text/javascript"> function formatQuizProgram2(matrix) {
if (matrixUID(matrix) != matrixUID(3, 3)) return false;
var result = true;
result = result && (equals(matrix[0][0], 3));
result = result && (equals(matrix[0][1], -9));
result = result && (equals(matrix[0][2], 16));
result = result && (equals(matrix[1][0], -5));
result = result && (equals(matrix[1][1], 5));
result = result && (equals(matrix[1][2], 17));
result = result && (equals(matrix[2][0], -14));
result = result && (equals(matrix[2][1], 20));
result = result && (equals(matrix[2][2], -4));
return result;
}
</script> </span> </div></div>
</div><div style="margin-top:20px"></div><span id="env14.3"></span><div class="exercise" data-count="14.3">
<div id="quiz3"> <div class="row quizquestion">
Use the following matrices to compute <span class="math"></span><script type="math/tex">\textbf{C}(\textbf{A}-\textbf{B})</script>.<div style="margin-top:20px"></div><span class="math"></span><script type="math/tex">
\textbf{A} =
\begin{pmatrix}
2 & 5 & -4 \\
4 & 0 & \phantom{-}9
\end{pmatrix}
</script>, <span class="math"></span><script type="math/tex">
\textbf{B} =
\begin{pmatrix}
1 & 6 & -4 \\
3 & 7 & \phantom{-}0
\end{pmatrix}
</script>
and
<span class="math"></span><script type="math/tex">
\textbf{C} =
\begin{pmatrix}
4 & -6 \\
2 & \phantom{-}1 \\
4 & -7
\end{pmatrix}
</script>
</div> <div class="row formatquizanswer"> <b>Your answer:</b> It is a <select class="formatquiz-type" onchange="formatQuizChangeType(3)"> <option value="none" selected></option> <option value="scalar">Scalar</option> <option value="columnvector">Column vector</option> <option value="rowvector">Row vector</option> <option value="matrix">Matrix</option> </select> <span class="formatquiz-scalar hidden"> which is </span> <span class="formatquiz-vector hidden"> with <select onchange="formatQuizChangeMatrixSize(3)"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> entries </span> <span class="formatquiz-matrix hidden"> with dimensions <select onchange="formatQuizChangeMatrixSize(3)" class="matrixrows"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> x <select onchange="formatQuizChangeMatrixSize(3)" class="matrixcolumns"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> </span> <span class="formatquiz-numbers hidden"> given by <table class="number-table"> </table> <input type="button" value="Check" onclick="formatQuizCheckAnswer(3, formatQuizProgram3)"> <span class="result"> <span class="number-errors"> </span> <span class="correct hidden"> <b>Correct!</b> </span> <span class="wrong hidden"> <b>Wrong.</b> </span> </span> <script type="text/javascript"> function formatQuizProgram3(matrix) {
if (matrixUID(matrix) != matrixUID(3, 3)) return false;
var result = true;
result = result && (equals(matrix[0][0], -2));
result = result && (equals(matrix[0][1], 38));
result = result && (equals(matrix[0][2], -54));
result = result && (equals(matrix[1][0], 3));
result = result && (equals(matrix[1][1], -9));
result = result && (equals(matrix[1][2], 9));
result = result && (equals(matrix[2][0], -3));
result = result && (equals(matrix[2][1], 45));
result = result && (equals(matrix[2][2], -63));
return result;
}
</script> </span> </div></div>
</div><div style="margin-top:20px"></div><span id="env14.4"></span><div class="exercise" data-count="14.4">
(Previous exam problem)<div style="margin-top:20px"></div>Let
<span class="math"></span><script type="math/tex"> \textbf{A} =
\begin{pmatrix}
5 & 3 \\
0 & 5
\end{pmatrix}</script>,
<span class="math"></span><script type="math/tex"> \textbf{B} =
\begin{pmatrix}
-8 & 0 & 7 \\
\phantom{-}1 & 3 & 2
\end{pmatrix}</script>,
and
<span class="math"></span><script type="math/tex">
\textbf{C} =
\begin{pmatrix}
1 & 0 \\
0 & 3 \\
7 & 1
\end{pmatrix}
</script><div style="margin-top:20px"></div>Show that <span class="math"></span><script type="math/tex">(\textbf{AB})\textbf{C} = \textbf{A}(\textbf{BC})</script>.
</div><div style="margin-top:20px"></div><span id="env14.5"></span><div class="exercise" data-count="14.5">
<div id="quiz4"> <div class="row quizquestion">
If <b>A</b> is a square matrix, then <span class="math"></span><script type="math/tex">\textbf{A}^2=\textbf{AA}</script>.
Let: <span class="math"></span><script type="math/tex">
\textbf{A} =
\begin{pmatrix}
8 & 4 \\
0 & 2
\end{pmatrix}
</script>.
Find <span class="math"></span><script type="math/tex">\textbf{A}^2</script>.
</div> <div class="row formatquizanswer"> <b>Your answer:</b> It is a <select class="formatquiz-type" onchange="formatQuizChangeType(4)"> <option value="none" selected></option> <option value="scalar">Scalar</option> <option value="columnvector">Column vector</option> <option value="rowvector">Row vector</option> <option value="matrix">Matrix</option> </select> <span class="formatquiz-scalar hidden"> which is </span> <span class="formatquiz-vector hidden"> with <select onchange="formatQuizChangeMatrixSize(4)"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> entries </span> <span class="formatquiz-matrix hidden"> with dimensions <select onchange="formatQuizChangeMatrixSize(4)" class="matrixrows"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> x <select onchange="formatQuizChangeMatrixSize(4)" class="matrixcolumns"> <option value="none" selected></option> <option value="1">1</option> <option value="2">2</option> <option value="3">3</option> <option value="4">4</option> </select> </span> <span class="formatquiz-numbers hidden"> given by <table class="number-table"> </table> <input type="button" value="Check" onclick="formatQuizCheckAnswer(4, formatQuizProgram4)"> <span class="result"> <span class="number-errors"> </span> <span class="correct hidden"> <b>Correct!</b> </span> <span class="wrong hidden"> <b>Wrong.</b> </span> </span> <script type="text/javascript"> function formatQuizProgram4(matrix) {
if (matrixUID(matrix) != matrixUID(2, 2)) return false;
var result = true;
result = result && (equals(matrix[0][0], 64));
result = result && (equals(matrix[0][1], 40));
result = result && (equals(matrix[1][0], 0));
result = result && (equals(matrix[1][1], 4));
return result;
}
</script> </span> </div></div>
</div><div style="margin-top:20px"></div><span id="env14.6"></span><div class="exercise" data-count="14.6">
Find the transpose of the following matrices:
<span class="math"></span><script type="math/tex">
\textbf{A} =
\begin{pmatrix}
3 & -8 \\
1 & \phantom{-}6 \\
4 & -9
\end{pmatrix}
</script>, <span class="math"></span><script type="math/tex"> \textbf{B} =
\begin{pmatrix}
0 \\
1 \\
1 \\
7
\end{pmatrix}</script>, and
<span class="math"></span><script type="math/tex">
\textbf{C} =
\begin{pmatrix}
2 & -3 & 5 & 9
\end{pmatrix}.
</script>
</div><div style="margin-top:20px"></div><span id="env14.7"></span><div class="exercise" data-count="14.7">
Show that the matrices
<span class="math"></span><script type="math/tex">
\textbf{A} =
\begin{pmatrix}
3 & \phantom{-}2 & 3 \\
2 & -1 & 1 \\
3 & \phantom{-}1 & 0
\end{pmatrix}
</script> and
<span class="math"></span><script type="math/tex">
\textbf{B} =
\begin{pmatrix}
0 & 4 & 8 \\
4 & 0 & 13 \\
8 & 13 & 0
\end{pmatrix}
</script> are both symmetric.
</div><div style="margin-top:20px"></div><span id="env14.8"></span><div class="exercise" data-count="14.8">
Denmark, Norway and Sweden have a significant trade with each other. All countries buy goods and services from each other (import) and sell goods and services to each other (export).<div style="margin-top:20px"></div>Their trade can be expressed in terms of a matrix <span class="math"></span><script type="math/tex">\textbf{A}</script>, where
<span class="math"></span><script type="math/tex">
\textbf{A} =
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}.
</script><div style="margin-top:20px"></div>Here, <span class="math"></span><script type="math/tex">a_{ij}</script> quantifies how much country <span class="math"></span><script type="math/tex">i</script> exports to country <span class="math"></span><script type="math/tex">j</script> in billions of kr. For example, if <span class="math"></span><script type="math/tex">a_{13}=5</script>, then country 1 exports goods and services worth 5 billion kr to country 3.<div style="margin-top:20px"></div><ol class="number"><li id="ite14.14"> What are the values of the diagonal elements (where <span class="math"></span><script type="math/tex">i=j</script>) in <span class="math"></span><script type="math/tex">\textbf{A}</script>?
</li><li id="ite14.15"> Is it a fair assumption to make that <span class="math"></span><script type="math/tex">\textbf{A}</script> is symmetric? Explain your answer.
</li><li id="ite14.16"> The minister of trade in country 1 requests a column vector with three elements where element number <span class="math"></span><script type="math/tex">i</script> indicates the total value of the trade between country 1 and country <span class="math"></span><script type="math/tex">i</script> (both import and export).
Which of the expressions below will result in such a vector? Explain your answer.
<ul><li> <span class="math"></span><script type="math/tex">2\textbf{A}</script>
</li><li> <span class="math"></span><script type="math/tex">\textbf{A}'</script>
</li><li> <span class="math"></span><script type="math/tex">(\textbf{A} + \textbf{A}')
\begin{pmatrix}
1 \\
0 \\
0
\end{pmatrix}
</script>
</li><li> <span class="math"></span><script type="math/tex">(\textbf{A} + \textbf{A}') </script>
</li><li> <span class="math"></span><script type="math/tex">\textbf{A}\textbf{A}' </script>
</li><li> <span class="math"></span><script type="math/tex">(\textbf{A} + \textbf{A}')
\begin{pmatrix}
0 \\
1 \\
1
\end{pmatrix}
</script>
</li></ul>
</li></ol>
</div><div style="margin-top:20px"></div><span id="env14.9"></span><div class="exercise" data-count="14.9">
Find the value of the determinant:
<span class="math"></span><script type="math/tex">
\begin{vmatrix}
4 & 6 \\
2 & 0
\end{vmatrix}
</script>
</div><div style="margin-top:20px"></div><span id="env14.10"></span><div class="exercise" data-count="14.10">
Find the value of the determinant:
<span class="math"></span><script type="math/tex">
\begin{vmatrix}
8 & 9 \\
5 & 4
\end{vmatrix}
</script>
</div><div style="margin-top:20px"></div><span id="env14.11"></span><div class="exercise" data-count="14.11">
Consider the following three systems of two equations with two unknowns:
<span id="equ14.1"></span><div class="math"></div><script type="math/tex; mode=display">\begin{aligned}
2x + 3y & = 3 \\
x - y & = - 1
\end{aligned}\tag{14.1}</script>
<span id="equ14.2"></span><div class="math"></div><script type="math/tex; mode=display">\begin{aligned}
3x + 2y + 1 & = 0 \\
-\tfrac{3}{2}x - y & = - 2
\end{aligned}\tag{14.2}</script>
<span id="equ14.3"></span><div class="math"></div><script type="math/tex; mode=display">\begin{aligned}
y - \tfrac{2}{3}x & = 2 \\
4x - 6y + 12 & = 0
\end{aligned}\tag{14.3}</script>
<ol class="number"><li id="ite14.17"> Write each of the three systems of equations in matrix form <span class="math"></span><script type="math/tex">\mathbf{Ax} = \mathbf{b}</script>.
</li><li id="ite14.18"> Calculate <span class="math"></span><script type="math/tex">\vert\mathbf{A}\vert</script> for each of the three systems of equations.
</li><li id="ite14.19"> Indicate which of the plots below corresponds to each of the three systems of equations.
</li><li id="ite14.20"> How many solutions are there to each of the three systems of equations?
</li></ol>
<div class="centerimg"><img src="img/week13/two_equations.png" width="80%"></div>
</div><div style="margin-top:20px"></div><span id="env14.12"></span><div class="exercise" data-count="14.12">
Use Cramer's rule to solve the system of equations, if possible:
<div class="math"></div><script type="math/tex; mode=display">\begin{aligned}
3x-8y & = 8 \\
x-2y & = 5
\end{aligned}</script>
</div><div style="margin-top:20px"></div><span id="env14.13"></span><div class="exercise" data-count="14.13">
Use Cramer's rule to solve the system of equations, if possible:
<div class="math"></div><script type="math/tex; mode=display">\begin{aligned}
x+3y & = 1 \\
3x-2y & = 14
\end{aligned}</script>
</div><div style="margin-top:20px"></div><span id="env14.14"></span><div class="exercise" data-count="14.14">
(Previous exam problem)
Let <span class="math"></span><script type="math/tex">a</script> and <span class="math"></span><script type="math/tex">b</script> be parameters. Use Cramer's rule to solve the following system of equations for the two unknown variables, <span class="math"></span><script type="math/tex">x</script> and
<span class="math"></span><script type="math/tex">y</script>:
<div class="math"></div><script type="math/tex; mode=display">\begin{aligned}
ax-by & = 1 \\
bx+ay & = 2
\end{aligned}</script>
</div><div style="margin-top:20px"></div><span id="env14.15"></span><div class="exercise" data-count="14.15">
(Previous exam problem)<div style="margin-top:20px"></div>Three matrices <span class="math"></span><script type="math/tex">\mathbf{A}</script>, <span class="math"></span><script type="math/tex">\mathbf{B}</script> and <span class="math"></span><script type="math/tex">\mathbf{C}</script> are given by
<div class="math"></div><script type="math/tex; mode=display">
\mathbf{A} = \left(\begin{array}{rr}
0 & -1 \\
1 & 0
\end{array}\right), \qquad
\mathbf{B} = \left(\begin{array}{rr}
-1 & 0 \\
0 & -1
\end{array}\right), \qquad
\mathbf{C} = \left(\begin{array}{rr}
0 & 1 \\
-1 & 0
\end{array}\right).
</script>
<ol class="alpha"><li id="ite14.21"> Show that <span class="math"></span><script type="math/tex">\mathbf{AB} = \mathbf{C}</script> and <span class="math"></span><script type="math/tex">\mathbf{BC} = \mathbf{A}</script>.
</li><li id="ite14.22"> Calculate the matrix product <span class="math"></span><script type="math/tex">\mathbf{CA}</script>.
</li></ol><div style="margin-top:20px"></div>The matrices <span class="math"></span><script type="math/tex">\mathbf{A}</script>, <span class="math"></span><script type="math/tex">\mathbf{B}</script> and <span class="math"></span><script type="math/tex">\mathbf{C}</script> are special cases of the general form
<div class="math"></div><script type="math/tex; mode=display">
\mathbf{D} = \left(\begin{array}{rr}
{\alpha} & {\beta} \\
-{\beta} & {\alpha}
\end{array}\right)
</script>
where <span class="math"></span><script type="math/tex">\alpha</script> and <span class="math"></span><script type="math/tex">\beta</script> are scalar constants (real numbers).<div style="margin-top:20px"></div><ol class="alpha" start = "3"><li id="ite14.23"> Find the values of <span class="math"></span><script type="math/tex">\alpha</script> and <span class="math"></span><script type="math/tex">\beta</script>, that ensure that <span class="math"></span><script type="math/tex">\mathbf{D}</script> satisfies the equation <span class="math"></span><script type="math/tex">\mathbf{D}^2 = \mathbf{I}</script>, where <span class="math"></span><script type="math/tex"> \mathbf{I}</script> is the identity matrix.
</li></ol>
</div><div style="margin-top:20px"></div><span id="sec14.4"></span><h2 id="988cdc9c-6611-42fc-b4eb-370b70fb44c1">14.4 Feedback</h2>
We would like to know which exercises helped you the most with <em>understanding</em> the mathematics and with your mathematical <em>skills</em>.<div style="margin-top:20px"></div>Examples of mathematical <em>understanding</em> might be an aha! experience of insight, understanding a concept by seeing it in different ways, or by thinking about why an answer was correct or incorrect.
Examples of mathematical <em>skills</em> might be the ability to make a new kind of calculation or analysis, or to become faster and more confident in applying a method.<div style="margin-top:20px"></div>Your answers are anonymous.<div style="margin-top:20px"></div>
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