<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>7.2.7.2 Exercise 2</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Let \(A = {@A@}\) and \(B = {@B@}\).</p>
<ol class="HELM_parts">
<li>Is it true that \(AB=BA\)? [[input:ans1]][[validation:ans1]][[feedback:prt1]]
</li><li> GIven that \(AB = \begin{bmatrix}\cos{z} &amp; \sin{z} \\ -\sin{z} &amp; \cos{z}\end{bmatrix}\), what is \(z\) in terms of \(x\) and \(y\)? [[input:ans2]][[validation:ans2]][[feedback:prt2]]<p></p></li></ol>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>We are given the matrices
    \(A = {@A@}\)
    and
    \(B = {@B@}\).</p>

<ol class="HELM_parts">
    <li>
        <p>First, we are asked whether \(AB = BA\), that is, whether \(A\) and \(B\) commute. In order to answer this question, we must find what \(AB\) and \(BA\) are.</p>
        \[\begin{aligned}
        AB &amp;= {@ABsteps@} \\
        &amp;= {@AB@}
        \end{aligned}\]
        \[\begin{aligned}
        BA &amp;= {@BAsteps@} \\
        &amp;= {@BA@}
        \end{aligned}\]
        <p>Checking each entry at a time, we see that \(AB\) is in fact equal to \(BA\).</p>
    </li>
    <li>
        <p>We already computed the value of \(AB\) in part (a). However, it was not fully simplified.</p>
        <p>Recall the trigonometric identities for the sum of angles:</p>
        \[\begin{aligned}
        \sin(x + y) &amp;= \sin x \cos y + \sin y \cos x \\
        \cos(x + y) &amp;= \cos x \cos y - \sin x \sin y
        \end{aligned}\]
        <p>Using these, we can rewrite our expression for \(AB\) as</p>
        \[{@ABfinal@}\]
        <p>Therefore, the value of \(z\) we are asked for is \(x + y\).</p>
    </li>
</ol>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020120600</text>
    </stackversion>
    <questionvariables>
      <text><![CDATA[A: matrix([cos(x), sin(x)], [-sin(x), cos(x)]);
B: ev(A, x=y);
ta : x + y;

/* Needed for solution */
AT: transpose(A);
BT: transpose(B);
ABsteps: zeromatrix(matrix_size(A)[1], matrix_size(B)[2]);
BAsteps: zeromatrix(matrix_size(B)[1], matrix_size(A)[2]);

simp: false;
for a: 1 thru matrix_size(ABsteps)[1] do
    for b: 1 thru matrix_size(ABsteps)[2] do
        ABsteps[ev(a,simp), ev(b,simp)]: apply("+", zip_with("*", A[ev(a,simp)], BT[ev(b,simp)]));
AB: ev(ABsteps, simp);

for a: 1 thru matrix_size(BAsteps)[1] do
    for b: 1 thru matrix_size(BAsteps)[2] do
        BAsteps[ev(a,simp), ev(b,simp)]: apply("+", zip_with("*", B[ev(a,simp)], AT[ev(b,simp)]));
BA: ev(BAsteps, simp);

simp: true;
ABfinal: trigreduce(A.B);]]></text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\(\theta+\phi\)</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>0</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>none</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans1</name>
      <type>dropdown</type>
      <tans>[[true,true,Yes],[false,false,No]]</tans>
      <boxsize>4</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
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      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>1</requirelowestterms>
      <checkanswertype>1</checkanswertype>
      <mustverify>0</mustverify>
      <showvalidation>0</showvalidation>
      <options/>
    </input>
    <input>
      <name>ans2</name>
      <type>algebraic</type>
      <tans>x+y</tans>
      <boxsize>10</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords>acos, asin</forbidwords>
      <allowwords>theta, phi</allowwords>
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        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>true</tans>
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        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans2</sans>
        <tans>ta</tans>
        <testoptions/>
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      </node>
    </prt>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>true</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta</value>
      </testinput>
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        <name>ans1</name>
        <value>first(mcq_incorrect([[true,true,Yes],[false,false,No]]))</value>
      </testinput>
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        <value>x-y</value>
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      </expected>
    </qtest>
  </question>
</quiz>
