<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>Matrices_Solve_Equation_2</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>If \[{\bf A}={@matrixa@}\]</p>
<p>find a non-zero 2 by 2 matrix \({\bf X}\) such that</p>
<p> \[{\bf A}\cdot{\bf X} = {\bf O}_{2x2}\]</p>
<p> </p>
<p>\({\bf O}_{2x2}\) is the 2 by 2 zero matrix.</p>
<p> [[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>Let  \({\bf X}\) = \({@matrixwxyz@}\)</p>
<p>\[{\bf A}\cdot{\bf X} = {@matrixa@}.{@matrixwxyz@}={@allzeros@}\]</p>
<p>Therefore<br />\[{@matrixa[1,1]*w+matrixa[1,2]*y@} = 0 \mbox{ ,            } {@matrixa[2,1]*w+matrixa[2,2]*y@} = 0\] \[{@matrixa[1,1]*x+matrixa[1,2]*z@} = 0  \mbox{ ,            }  {@matrixa[2,1]*x+matrixa[2,2]*z@} = 0\]</p>
<p>The 2 equations in \(w\) and \(y\) are linearly dependent as are the 2 equations in \(x\) and \(z\).</p>
<p>Therefore there is no unique solution but \[w={@-matrixa[1,2]*y/matrixa[1,1]@} \mbox{     and       } x={@-matrixa[1,2]*z/matrixa[1,1]@} \]</p>
<p>So \({\bf X}\) could be \({@matrixx@}\)</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text/>
    </stackversion>
    <questionvariables>
      <text>n1:-(rand(9)+1);
 n3:rand(4)+2;
 n2:-n1*n3+1;
 matrixa:matrix([n1,n2],[n1/n3,n2/n3]);
 matrixx:matrix([n2,-n2/n3],[-n1,n1/n3]);
 allzeros: zeromatrix(2,2);
 matrixwxyz:matrix([w,x],[y,z]);
 wronganswer1:matrixa;</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\({\bf A} = {@matrixa@}, {\bf A}\cdot{\bf X} = {\bf O}_{2x2}, {\bf X} ={@matrixx@}\)</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>
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    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
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    <input>
      <name>ans1</name>
      <type>matrix</type>
      <tans>matrixx</tans>
      <boxsize>20</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords>determinant,invert</forbidwords>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>1</requirelowestterms>
      <checkanswertype>1</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>1</showvalidation>
      <options/>
    </input>
    <prt>
      <name>Result</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text>sanswer:matrixa.ans1;</text>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>allzeros</tans>
        <testoptions/>
        <quiet>1</quiet>
        <truescoremode>=</truescoremode>
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        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-0-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Your answer must be a non-zero 2 by 2 matrix.</p>]]></text>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty>0</falsepenalty>
        <falsenextnode>1</falsenextnode>
        <falseanswernote>Result-0-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>1</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>sanswer</sans>
        <tans>allzeros</tans>
        <testoptions/>
        <quiet>1</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty>0</truepenalty>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty>0</falsepenalty>
        <falsenextnode>2</falsenextnode>
        <falseanswernote>Result-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>2</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>wronganswer1</tans>
        <testoptions/>
        <quiet>1</quiet>
        <truescoremode>=</truescoremode>
        <truescore>0.0000000</truescore>
        <truepenalty>0</truepenalty>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-2-T</trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>You have entered matrix \({\bf A}\).</p>]]></text>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty>0</falsepenalty>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>Result-2-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
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    <deployedseed>171825923</deployedseed>
    <deployedseed>99093707</deployedseed>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>allzeros</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-0-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>matrixx</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer1</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-2-T</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
