// Numbas version: exam_results_page_options
{"name": "Eigenvalues & eigenvectors of a 2x2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "variables": {"a12": {"name": "a12", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "k*(a11-c1)"}, "lambda2": {"name": "lambda2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "max({c1},{a11}+{a22}-{c1})"}, "c1": {"name": "c1", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(14..20#1)"}, "k": {"name": "k", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(1..6#1)"}, "lambda1": {"name": "lambda1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "min({c1},{a11}+{a22}-{c1})"}, "a11": {"name": "a11", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(1..10#1)"}, "a22": {"name": "a22", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{k}*{a21}+{c1}"}, "a21": {"name": "a21", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(1..5#1)"}}, "metadata": {"description": "<p>This question concerns the evaluation of the eigenvalues and corresponding eigenvectors of a 2x2 matrix.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "name": "Eigenvalues & eigenvectors of a 2x2", "parts": [{"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "lambda1", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": 0, "correctAnswerFraction": false, "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "lambda1", "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "marks": 1, "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "lambda2", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": 0, "correctAnswerFraction": false, "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "lambda2", "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "marks": 1, "correctAnswerStyle": "plain"}], "prompt": "<p>Calculate the eigenvalues of the matrix A</p>\n<p>\\(\\lambda_1\\) is the lesser of the two eigenvalues and&nbsp;\\(\\lambda_2\\) is the greater of the two eigenvalues;</p>\n<p style=\"padding-left: 30px;\">\\(\\lambda_1\\) =&nbsp;[[0]]</p>\n<p style=\"padding-left: 30px;\">\\(\\lambda_2\\) =&nbsp;[[1]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}, {"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{lambda1}-{a22}", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": 0, "correctAnswerFraction": false, "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{lambda1}-{a22}", "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "marks": 1, "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "{lambda2}-{a22}", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": 0, "correctAnswerFraction": false, "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{lambda2}-{a22}", "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "marks": 1, "correctAnswerStyle": "plain"}], "prompt": "<p>For the lesser eigenvalue \\(\\lambda_1\\) the corresponding eigenvector is \\(v_1=\\begin{pmatrix}x\\\\ \\var{a21}\\\\ \\end{pmatrix}\\)</p>\n<p>Enter the value for \\(x=\\)&nbsp;[[0]]</p>\n<p>For the greater eigenvalue \\(\\lambda_2\\) the corresponding eigenvector is \\(v_1=\\begin{pmatrix}x\\\\ \\var{a21}\\\\ \\end{pmatrix}\\)</p>\n<p>Enter the value for \\(x=\\)&nbsp;[[1]]</p>\n<p></p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}], "rulesets": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "extensions": [], "statement": "<p>Given the matrix</p>\n<p style=\"padding-left: 60px;\">\\(A =\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\)</p>", "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a11", "c1", "a12", "k", "a21", "a22", "lambda1", "lambda2"], "advice": "<p>The eigenvalues of a matrix are the values of \\(\\lambda\\) that satisfy the relation</p>\n<p style=\"padding-left: 30px;\">\\(|A-\\lambda I| = 0\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{vmatrix} \\var{a11}-\\lambda&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}-\\lambda\\\\ \\end{vmatrix}=0\\)</p>\n<p>This gives:</p>\n<p style=\"padding-left: 30px;\">\\((\\var{a11}-\\lambda)*(\\var{a22}-\\lambda)-(\\var{a12})*(\\var{a21})=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\lambda^2-\\simplify{{a11}+{a22}}\\lambda+\\simplify{{a11}*{a22}-{a21}*{a12}}=0\\)</p>\n<p>This can be solved using factorisation or by formula to give:</p>\n<p style=\"padding-left: 30px;\">\\(\\lambda =\\var{lambda1}\\) and&nbsp;\\(\\lambda =\\var{lambda2}\\)</p>\n<p>An eigenvector \\(v=\\begin{pmatrix} x\\\\ y\\\\ \\end{pmatrix}\\) corresponding to an eigenvalue \\(\\lambda\\) must satisfy the relation: &nbsp;\\((A-\\lambda I)v = 0\\)</p>\n<p>so for \\(\\lambda=\\var{lambda1}\\) &nbsp;</p>\n<p style=\"padding-left: 60px;\">\\(\\begin{pmatrix} \\simplify{{a11}-{lambda1}}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\simplify{{a22}-{lambda1}}\\\\ \\end{pmatrix}\\begin{pmatrix} x\\\\ \\var{a21}\\\\ \\end{pmatrix}=0\\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</p>\n<p>thus</p>\n<p style=\"padding-left: 60px;\">\\(\\var{a21}x+\\simplify{{a22}-{lambda1}}*\\var{a21}=0\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\var{a21}x=-\\simplify{({a22}-{lambda1})*{a21}}\\)</p>\n<p style=\"padding-left: 60px;\">\\(x=-\\simplify{({a22}-{lambda1})}\\)</p>\n<p></p>", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}