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    <name>
      <text>7.3.4.1 Exercise 1</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>

<p>In this exercise you will use Laplace expansion along the 1st row to determine the determinant of the following matrix:</p><div>\(A={@matrixa@}\)</div>

<p>Let the entries in the first row be denoted \(a_{1,1}\), \(a_{1,2}\) and \(a_{1,3}\) in the usual way. First list the cofactors \(A_{i,j}\) of these entries:</p>
<p>\(A_{1,1}=\)[[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>
<p>\(A_{1,2}=\)[[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>
<p>\(A_{1,3}=\)[[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>

<p>Use these values to evaluate the determinant of \(A\): [[input:ans4]] [[validation:ans4]] [[feedback:prt4]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text>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.

For \(a_{1,1}\), the sign is \(+\), and the minor is \(\det{@matrix([AA[2,2], AA[2,3]], [AA[3,2], AA[3,3]])@}\). Hence the cofactor is \({@ta1@}.\)

For \(a_{1,2}\), the sign is \(-\), and the minor is \(\det{@matrix([AA[2,1], AA[2,3]], [AA[3,1], AA[3,3]])@}\). Hence the cofactor is \({@ta2@}.\)

For \(a_{1,3}\), the sign is \(+\) again, and the minor is \(\det{@matrix([AA[2,1], AA[2,2]], [AA[3,1], AA[3,2]])@}\). Hence the cofactor is \({@ta3@}.\)

The determinant of \(A\) is now given by the formula:
\[\det(A) = {@CC@} = {@DD@}\]</text>
    </generalfeedback>
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    <stackversion>
      <text>2021072200</text>
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    <questionvariables>
      <text><![CDATA[matrixa: matrix([rand([-5,-4,-3,-2,-1,1,2,3,4,5]), rand([-5,-4,-3,-2,-1,1,2,3,4,5]), rand([-5,-4,-3,-2,-1,1,2,3,4,5])], [rand([-5,-4,-3,-2,-1,1,2,3,4,5]), rand([-5,-4,-3,-2,-1,1,2,3,4,5]), rand([-5,-4,-3,-2,-1,1,2,3,4,5])], [rand([-5,-4,-3,-2,-1,1,2,3,4,5]), rand([-5,-4,-3,-2,-1,1,2,3,4,5]), rand([-5,-4,-3,-2,-1,1,2,3,4,5])]);
matrixa: ev(matrixa,simp);

AA: matrixa;
MM11: matrix([AA[2,2], AA[2,3]], [AA[3,2], AA[3,3]]);
MM12: matrix([AA[2,1], AA[2,3]], [AA[3,1], AA[3,3]]);
MM13: matrix([AA[2,1], AA[2,2]], [AA[3,1], AA[3,2]]);

ta1: ev(determinant(MM11),simp)
ta2: ev(-determinant(MM12),simp)
ta3: ev(determinant(MM13),simp)
ta4: ev(determinant(matrixa),simp);

minorlist: [determinant(MM11),-determinant(MM12),determinant(MM13)];
simp: false;
CC: apply("+",zip_with("*",AA[1],minorlist));
DD: ev(CC,simp);]]></text>
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      <text/>
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    <questionnote format="moodle_auto_format">
      <text>\( A = {@AA@}, \, A_{1,1} ={@ta1@}, \, A_{1,2} = {@ta2@}, \, A_{1,3}={@ta3@}, \, detA=  {@ta4@} \)</text>
    </questionnote>
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    <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>
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      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
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    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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