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<quiz>
  <question type="stack">
    <name>
      <text>7.3.4.3 Exercise 3</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>

<p>Find the cofactors of \(x,y,z\) in the following determinant:</p>
\[\begin{vmatrix} {@AA[1,1]@} &amp; {@AA[1,2]@} &amp; {@AA[1,3]@} \\ {@AA[2,1]@} &amp; {@AA[2,2]@} &amp; {@AA[2,3]@} \\ x &amp; y &amp; z \end{vmatrix}\]

<p>Cofactor of \(x\): [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>
<p>Cofactor of \(y\): [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>
<p>Cofactor of \(z\): [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[To obtain the cofactor of an element of a \(3 \times 3\) matrix we multiply the minor of that element by the corresponding sign from the \(3 \times 3\) array of signs.<br>We are given the determinant: \(\begin{vmatrix} {@AA[1,1]@} &amp; {@AA[1,2]@} &amp; {@AA[1,3]@} \\ {@AA[2,1]@} &amp; {@AA[2,2]@} &amp; {@AA[2,3]@} \\ x &amp; y &amp; z \end{vmatrix}.\)<br><p>For \(x\), the sign is \(+\), and the minor is \(\det {@MMx@}\). Hence the cofactor is \({@ta1@}.\)</p><p>For \(y\), the sign is \(-\), and the minor is \(\det {@MMy@}\). Hence the cofactor is \({@ta2@}.\)</p><p>For \(z\), the sign is \(+\) again, and the minor is \(\det {@MMz@}\). Hence the cofactor is \({@ta3@}.\)</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2021072200</text>
    </stackversion>
    <questionvariables>
      <text>matrixa: matrix([rand_with_step(-5,5,1), rand_with_step(-5,5,1), rand_with_step(-5,5,1)],[rand_with_step(-5,5,1), rand_with_step(-5,5,1), rand_with_step(-5,5,1)])
matrixa: ev(matrixa,simp);

AA: matrixa;
MMx: matrix([AA[1,2], AA[1,3]], [AA[2,2], AA[2,3]]);
MMy: matrix([AA[1,1], AA[1,3]], [AA[2,1], AA[2,3]]);
MMz: matrix([AA[1,1], AA[1,2]], [AA[2,1], AA[2,2]]);

ta1: ev(determinant(MMx),simp)
ta2: ev(-determinant(MMy),simp)
ta3: ev(determinant(MMz),simp)</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\({@AA@}, \, x:{@ta1@}, \, y:{@ta2@}, \, z:{@ta3@}\)</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</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>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>algebraic</type>
      <tans>ta1</tans>
      <boxsize>4</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
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      <options/>
    </input>
    <input>
      <name>ans2</name>
      <type>algebraic</type>
      <tans>ta2</tans>
      <boxsize>4</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords>determinant</forbidwords>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
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      <checkanswertype>0</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <input>
      <name>ans3</name>
      <type>algebraic</type>
      <tans>ta3</tans>
      <boxsize>4</boxsize>
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      <value>1.0000000</value>
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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>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt2-1-T</expectedanswernote>
      </expected>
      <expected>
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        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt3-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</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>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-1-F</expectedanswernote>
      </expected>
      <expected>
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        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt2-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt3-1-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
