// Numbas version: exam_results_page_options
{"name": "Eigenvalues and Eigenvectors of a 2x2 Matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["test1", "a21", "a22", "b1", "mna", "mnb", "s2", "s1", "a12", "test", "a11", "that", "da", "a1", "c2", "c1", "mxb", "mxa", "a", "b", "tra", "n", "this", "s"], "name": "Eigenvalues and Eigenvectors of a 2x2 Matrix", "tags": ["diagonalising matrices", "eigenvalues", "eigenvalues of matrix", "eigenvectors of matrix", "matrix", "matrix eigenvalues", "test5"], "advice": "<h4>Matrix $A$</h4>\n<p>\\[A - \\lambda I_2 = \\begin{pmatrix} \\var{a11}-\\lambda &amp; \\var{a12}\\\\ \\var{a21} &amp; \\var{a22}-\\lambda \\end{pmatrix}\\]<br/>Hence the characteristic polynomial $p(\\lambda)$ is: \\[\\begin{eqnarray*} \\mathrm{det}\\left(A-\\lambda I_2 \\right)&amp;=&amp;\\simplify[zeroTerm]{({a11}-lambda)({a22}-lambda)-{a12}*{a21}}\\\\ &amp;=&amp; \\simplify[std]{lambda^2-{trA}*lambda+{dA}}\\\\ &amp;=&amp;\\simplify[std]{(lambda-{a})(lambda-{b})} \\end{eqnarray*} \\]<br/>We see that on solving $p(\\lambda)=0$ we get the eigenvalues:<br/>\\[\\lambda_1=\\var{mnA},\\;\\;\\;\\lambda_2=\\var{mxA}\\] <br/>Note: We could have found the characteristic polynomial by noting that for a 2 &times; 2 matrix $A$ then the characteristic polynomial is<br/>\\[\\lambda^2-\\mathrm{trace}(A)+\\mathrm{det}(A)\\]<br/>where $\\mathrm{trace}(A) = \\var{trA},\\;\\;\\;\\mathrm{det}(A)=\\var{dA}$</p>\n<h4>Finding the eigenvectors:</h4>\n<p>1. $\\lambda_1=\\var{mnA}$</p>\n<p>We have the eigenspace is given by all $v=(x,y)^\\mathrm{T}$ such that $(\\simplify{A-{mnA}}I_2)v=(0,0)^\\mathrm{T}$ i.e.</p>\n<p>\\[\\begin{pmatrix} \\var{a11-mnA}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22-mnA} \\end{pmatrix}\\begin{pmatrix} x \\\\ y \\end{pmatrix} =\\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}\\]</p>\n<p>This gives the two equations:</p>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{{a11-mnA}x + {a12}y}&amp;=&amp;0\\\\ \\simplify[std]{{a21}x + {a22-mnA}y}&amp;=&amp;0 \\end{eqnarray*} \\]<br/>There is only one equation here as we see that the equations are the same (one is a multiple of the other).</p>\n<p>So putting $x=1$ in the first equation we get $y_1=\\var{-s*(a11-mnA)}$</p>\n<p>Hence the eigenvector we want is \\[\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mnA)} \\end{pmatrix}\\]</p>\n<p>2. $\\lambda_2=\\var{mxA}$</p>\n<p>In this case we have the equations:</p>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{{a11-mxA}x + {a12}y}&amp;=&amp;0\\\\ \\simplify[std]{{a21}x + {a22-mxA}y}&amp;=&amp;0 \\end{eqnarray*} \\]</p>\n<p>Once again there is only one equation, so putting $x=1$ in the first equation we get $y_2=\\var{-s*(a11-mxA)}$</p>\n<p>Hence the eigenvector we want is \\[\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mxA)} \\end{pmatrix}\\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      \n      \n      <p>Find the eigenvalues of $A$.</p>\n      \n      \n      \n      <p>Let $a_1$ be the least eigenvalue of $A,\\;\\;\\; a_1=\\;\\;$[[0]]</p>\n      \n      \n      \n      <p>Let $a_2$ be the greatest eigenvalue of $A,\\;\\; a_2=\\;\\;$[[1]]</p>\n      \n      \n        \n      \n      \n        \n      ", "marks": 0, "gaps": [{"allowFractions": false, "scripts": {}, "maxValue": "{mnA}", "minValue": "{mnA}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "{mxA}", "minValue": "{mxA}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Find eigenvectors for $A$.</p>\n<p>Let $(1,y_1)^\\mathrm{T}$ be an eigenvector corresponding to the eigenvalue&nbsp;$a_1$.</p>\n<p>$y_1=$&nbsp;[[0]]</p>\n<p>Let $(1,y_2)^\\mathrm{T}$ be an eigenvector corresponding to the eigenvalue&nbsp;$a_2$.</p>\n<p>$y_2=$&nbsp;[[1]]</p>", "marks": 0, "gaps": [{"allowFractions": false, "scripts": {}, "maxValue": "{s*(mnA-a11)}", "minValue": "{s*(mnA-a11)}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "{s*(mxA-a11)}", "minValue": "{s*(mxA-a11)}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Find the eigenvalues and eigenvectors for the matrix $A$ where:<br/>\\[ A=\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22} \\end{pmatrix}\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"test1": {"definition": "random(-8..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "test1", "description": ""}, "a21": {"definition": "-s*(a*b-a11*a-a11*b+a11^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a21", "description": ""}, "a22": {"definition": "a+b-a11", "templateType": "anything", "group": "Ungrouped variables", "name": "a22", "description": ""}, "b1": {"definition": "if(c2=a1,a1+random(1..3),c2)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "mna": {"definition": "min(a,b)", "templateType": "anything", "group": "Ungrouped variables", "name": "mna", "description": ""}, "mnb": {"definition": "min(a1,b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "mnb", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "a12": {"definition": "s", "templateType": "anything", "group": "Ungrouped variables", "name": "a12", "description": ""}, "test": {"definition": "random(-8..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "test", "description": ""}, "a11": {"definition": "switch(test=a,this,test=b,this,test)", "templateType": "anything", "group": "Ungrouped variables", "name": "a11", "description": ""}, "that": {"definition": "if(mxB<0,mxB+random(1..6),mxB-random(1..6))", "templateType": "anything", "group": "Ungrouped variables", "name": "that", "description": ""}, "da": {"definition": "{a11*a22-a12*a21}", "templateType": "anything", "group": "Ungrouped variables", "name": "da", "description": ""}, "a1": {"definition": "random(-8..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "c2": {"definition": "random(-8..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}, "c1": {"definition": "random(-8..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "mxb": {"definition": "max(a1,b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "mxb", "description": ""}, "mxa": {"definition": "max(a,b)", "templateType": "anything", "group": "Ungrouped variables", "name": "mxa", "description": ""}, "a": {"definition": "random(-8..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "if(c1=a,a+random(1..3),c1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "tra": {"definition": "{a11+a22}", "templateType": "anything", "group": "Ungrouped variables", "name": "tra", "description": ""}, "n": {"definition": "random(4,5,6)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "this": {"definition": "if(mxA<0,mxA+random(1..6),mxA-random(1..6))", "templateType": "anything", "group": "Ungrouped variables", "name": "this", "description": ""}, "s": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}}, "metadata": {"notes": "<p><strong>21/04/13</strong> Based on Bill's \"Eigenvalues and eigenvectors\" combo question...</p>", "description": "<p>Find eigenvalues and eigenvectors of $A$ $2 \\times 2$ matrix.</p>", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Hayley Bishop", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/93/"}]}]}], "contributors": [{"name": "Hayley Bishop", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/93/"}]}