<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>7.4.2.4 Inverse of a 2 x 2 matrix</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find the inverses of the following matrices</p>
<p>\(A={@A@}\), \(B={@B@}\), \(C={@C@}\).</p>
<ol class="HELM_parts">
<li><div>\(A^{-1}=\) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</div></li>
<li><div>\(B^{-1}=\) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</div></li>
<li><div>\(C^{-1}=\) [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</div></li>
</ol>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>Recall that the inverse of a \(2 \times 2\) matrix
    \(A = {@matrixabcd@}\) is given by the formula</p>
\[A^{-1} = \frac{1}{ad - bc} {@matrixdbca@}.\]
<p>We apply this to each of the given matrices.</p>

<ol class="HELM_parts">
    <li><p>For \(A = {@A@}\), we have</p>
        \[A^{-1} = \frac{1}{{@determinant(A)@}} {@preInv(A)@} = {@ta1@}.\]
    </li>
    <li><p>For \(B = {@B@}\), we have</p>
        \[B^{-1} = \frac{1}{{@determinant(B)@}} {@preInv(B)@} = {@ta2@}.\]
    </li>
    <li><p>For \(C = {@C@}\), we have</p>
        \[C^{-1} = \frac{1}{{@determinant(C)@}} {@preInv(C)@} = {@ta3@}.\]
    </li>
</ol>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020120600</text>
    </stackversion>
    <questionvariables>
      <text>A:matrix([rand(4)+5,rand(4)+1],[rand(4)+1,rand(4)+5]);
B:matrix([rand(9)+1,0],[0,rand(9)+1]);

p:matrix([0,2],[2,2]); q:matrix([2,0],[2,2]); r:matrix([2,2],[0,2]); s:matrix([2,2],[2,0]);
t:matrix([2,0],[0,0]); u:matrix([0,2],[0,0]); v:matrix([0,0],[2,0]); w:matrix([0,0],[0,2]);

Ca:matrix([-1,-1],[-1,-1]);
Cb:[p,q,r,s,t,u,v,w];
C:Ca+rand(Cb);

ta1:invert(A);
ta2:invert(B);
ta3:invert(C);

/* Needed for answer */
preInv(A) := matrix([A[2,2], -A[1,2]], [-A[2,1], A[1,1]]);

/* Needed for feedback */
matrixabcd: matrix([a,b],[c,d]);
matrixdbca: matrix([d,-b],[-c,a]);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\[\bf A = {@A@}, B = {@B@}, C = {@C@}\] 
\[A^{-1} = {@ta1@}, B^{-1} = {@ta2@},  C^{-1} = {@ta3@}\]</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <text>Correct answer, well done.</text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text>Your answer is partially correct.</text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text>Incorrect answer.</text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</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>matrix</type>
      <tans>ta1</tans>
      <boxsize>5</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
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      <options/>
    </input>
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      <name>ans2</name>
      <type>matrix</type>
      <tans>ta2</tans>
      <boxsize>5</boxsize>
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      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <input>
      <name>ans3</name>
      <type>matrix</type>
      <tans>ta3</tans>
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      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <prt>
      <name>prt1</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text>wronganswer1: ev(-A,simp);
wronganswer2: zeromatrix(2,2);</text>
      </feedbackvariables>
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        <name>0</name>
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        <truefeedback format="html">
          <text><![CDATA[<p>It appears you have given the matrix \(-A\)! This is not the inverse matrix. Recall the inverse of a 2 \(\times\) 2 matrix \[{@matrixabcd@}\]&nbsp;is \[\frac{1}{ad-bc}{@matrixdbca@}.\] Please try part (a) again!</p>]]></text>
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          <text/>
        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt2</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
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        <text>wronganswer3: ev(-B,simp);</text>
      </feedbackvariables>
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      <node>
        <name>1</name>
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        <answertest>AlgEquiv</answertest>
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        <trueanswernote>prt2-2-T</trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>It appears you have given the matrix \(-B\)! This is not the inverse matrix. Recall the inverse of a 2 \(\times\) 2 matrix \[{@matrixabcd@}\]&nbsp;is \[\frac{1}{ad-bc}{@matrixdbca@}.\] Please try part (b) again!<br></p>]]></text>
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        <answertest>AlgEquiv</answertest>
        <sans>ans3</sans>
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          <text><![CDATA[<p>It appears you have given the matrix \(-C\)! This is not the inverse matrix. Recall the inverse of a 2 \(\times\) 2 matrix \[{@matrixabcd@}\]&nbsp;is \[\frac{1}{ad-bc}{@matrixdbca@}.\] Please try part (c) again!<br></p>]]></text>
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    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ta3</value>
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      <testcase>2</testcase>
      <description/>
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        <name>ans1</name>
        <value>ev(-A,simp)</value>
      </testinput>
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        <value>ev(-B,simp)</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ev(-C,simp)</value>
      </testinput>
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        <expectedpenalty>0.0000000</expectedpenalty>
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        <expectedpenalty>0.0000000</expectedpenalty>
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      </expected>
      <expected>
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        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt3-2-T</expectedanswernote>
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    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ev(ta1+1,simp)</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ev(ta2+1,simp)</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ev(ta3+1,simp)</value>
      </testinput>
      <expected>
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        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-2-F</expectedanswernote>
      </expected>
      <expected>
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        <expectedscore>0.0000000</expectedscore>
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      </expected>
      <expected>
        <name>prt3</name>
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      </expected>
    </qtest>
  </question>
</quiz>
