// Numbas version: exam_results_page_options
{"name": "Ch10 Q5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "extensions": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "<p>a)</p>\n<h3>Matrix $A$</h3>\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<p>b)</p>\n<h4>Finding the eigenvectors:</h4>\n<p>1. $\\lambda=\\var{mnA}$</p>\n<p>We have the eigenspace is given by all $v=(x,y)$ such that $(\\simplify{A-{mnA}}I_2)v=(0,0)$ 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. $\\lambda=\\var{mxA}$</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>", "statement": "<p>Find the eigenvalues and eigenvectors for the matrix $A$&nbsp;where:<br/>\\[ A=\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22} \\end{pmatrix}\\;\\;\\;\\;\\;\\]</p>", "variable_groups": [], "variables": {"a1": {"definition": "random(-8..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a1"}, "bn22": {"definition": "round(s*cn22/f)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "bn22"}, "cn21": {"definition": "mnB^n-mxB^n", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "cn21"}, "a11": {"definition": "switch(test=a,this,test=b,this,test)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a11"}, "test": {"definition": "random(-8..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "test"}, "b": {"definition": "if(c1=a,a+random(1..3),c1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b"}, "cn12": {"definition": "x1*x2*(mxB^n-mnB^n)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "cn12"}, "mna": {"definition": "min(a,b)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "mna"}, "bn21": {"definition": "round(s*cn21/f)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "bn21"}, "a12": {"definition": "s", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a12"}, "x2": {"definition": "s*(b22-mxB)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "x2"}, "s": {"definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s"}, "this": {"definition": "if(mxA<0,mxA+random(1..6),mxA-random(1..6))", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "this"}, "b22": {"definition": "a1+b1-b11", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b22"}, "a22": {"definition": "a+b-a11", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a22"}, "f": {"definition": "abs(x1-x2)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "f"}, "that": {"definition": "if(mxB<0,mxB+random(1..6),mxB-random(1..6))", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "that"}, "s2": {"definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s2"}, "x1": {"definition": "s*(b22-mnB)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "x1"}, "mxa": {"definition": "max(a,b)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "mxa"}, "a": {"definition": "random(-8..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a"}, "cn11": {"definition": "mnB^n*x1-mxB^n*x2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "cn11"}, "test1": {"definition": "random(-8..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "test1"}, "b1": {"definition": "if(c2=a1,a1+random(1..3),c2)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b1"}, "bn12": {"definition": "round(s*cn12/f)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "bn12"}, "da": {"definition": "{a11*a22-a12*a21}", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "da"}, "s1": {"definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s1"}, "cn22": {"definition": "mxB^n*x1-mnB^n*x2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "cn22"}, "n": {"definition": "random(4,5,6)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "n"}, "mxb": {"definition": "max(a1,b1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "mxb"}, "b11": {"definition": "switch(test1=a1,that,test1=b1,that,test1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b11"}, "b21": {"definition": "-s", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b21"}, "trb": {"definition": "{b11+b22}", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "trb"}, "mnb": {"definition": "min(a1,b1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "mnb"}, "c1": {"definition": "random(-8..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c1"}, "c2": {"definition": "random(-8..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c2"}, "db": {"definition": "{b11*b22-b12*b21}", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "db"}, "b12": {"definition": "s*(a1*b1-b11*a1-b11*b1+b11^2)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b12"}, "bn11": {"definition": "round(s*cn11/f)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "bn11"}, "tra": {"definition": "{a11+a22}", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "tra"}, "a21": {"definition": "-s*(a*b-a11*a-a11*b+a11^2)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a21"}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>$A,\\;B$ $2 \\times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.</p>"}, "ungrouped_variables": ["test1", "a21", "a22", "bn22", "bn21", "b22", "b21", "b1", "cn11", "mna", "mnb", "cn12", "s2", "s1", "a12", "b12", "b11", "test", "cn21", "cn22", "a11", "that", "db", "bn12", "bn11", "da", "a1", "x2", "c2", "c1", "x1", "mxb", "mxa", "a", "b", "f", "tra", "trb", "n", "this", "s"], "tags": [], "parts": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "<p>Find the eigenvalues of $A$.</p>\n<p>Let $a_1$ be the least eigenvalue of $A,\\;\\;\\; a_1=\\;\\;$[[0]]</p>\n<p>Let $a_2$ be the greatest eigenvalue of $A,\\;\\; a_2=\\;\\;$[[1]]</p>", "gaps": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "marks": 1, "correctAnswerFraction": false, "minValue": "{mnA}", "maxValue": "{mnA}", "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "marks": 1, "correctAnswerFraction": false, "minValue": "{mxA}", "maxValue": "{mxA}", "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false}], "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill"}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "<p>Find eigenvectors for $A$.</p>\n<p>Let $(1,y_1)$ be an eigenvector corresponding to $a_1,\\;\\;\\;\\;y_1=\\;\\;$[[0]]</p>\n<p>Let $(1,y_2)$ be an eigenvector corresponding to $a_2,\\;\\;\\;\\;y_2=\\;\\;$[[1]]</p>", "gaps": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "marks": 1, "correctAnswerFraction": false, "minValue": "{s*(mnA-a11)}", "maxValue": "{s*(mnA-a11)}", "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "marks": 1, "correctAnswerFraction": false, "minValue": "{s*(mxA-a11)}", "maxValue": "{s*(mxA-a11)}", "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false}], "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill"}], "name": "Ch10 Q5", "type": "question", "contributors": [{"name": "Alex Van den Hof", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1273/"}]}]}], "contributors": [{"name": "Alex Van den Hof", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1273/"}]}