// Numbas version: finer_feedback_settings
{"name": "Eigenvalues & eigenvectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p style=\"text-align: center;\"><iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/0HZ8Iy4e3eg?rel=0&amp;showinfo=0\" frameborder=\"0\" gesture=\"media\" allow=\"encrypted-media\" allowfullscreen=\"allowfullscreen\"></iframe></p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\"><iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/CPQRM14Jq1Y?rel=0&amp;showinfo=0\" frameborder=\"0\" gesture=\"media\" allow=\"encrypted-media\" allowfullscreen=\"allowfullscreen\"></iframe></p>\n<p></p>\n<p>Given the matrix</p>\n<p style=\"padding-left: 60px; text-align: center;\">\\(A =\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\)</p>", "parts": [{"scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "gaps": [{"maxValue": "lambda1", "precision": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "showPrecisionHint": false, "allowFractions": false, "minValue": "lambda1", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "type": "numberentry", "precisionType": "dp", "strictPrecision": false, "scripts": {}, "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "marks": 1}, {"maxValue": "lambda2", "precision": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "showPrecisionHint": false, "allowFractions": false, "minValue": "lambda2", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "type": "numberentry", "precisionType": "dp", "strictPrecision": false, "scripts": {}, "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "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>"}, {"scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "gaps": [{"maxValue": "{lambda1}-{a22}", "precision": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "showPrecisionHint": false, "allowFractions": false, "minValue": "{lambda1}-{a22}", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "type": "numberentry", "precisionType": "dp", "strictPrecision": false, "scripts": {}, "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "marks": 1}, {"maxValue": "{lambda2}-{a22}", "precision": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "showPrecisionHint": false, "allowFractions": false, "minValue": "{lambda2}-{a22}", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "type": "numberentry", "precisionType": "dp", "strictPrecision": false, "scripts": {}, "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>For the lesser eigenvalue \\(\\lambda_1\\) the corresponding eigenvector is \\(v_1=\\begin{pmatrix}x_1\\\\ \\var{a21}\\\\ \\end{pmatrix}\\)</p>\n<p>Enter the value for \\(x_1=\\)&nbsp;[[0]]</p>\n<p>For the greater eigenvalue \\(\\lambda_2\\) the corresponding eigenvector is \\(v_1=\\begin{pmatrix}x_2\\\\ \\var{a21}\\\\ \\end{pmatrix}\\)</p>\n<p>Enter the value for \\(x_2=\\)&nbsp;[[1]]</p>\n<p></p>"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>This question concerns the evaluation of the eigenvalues and corresponding eigenvectors of a 2x2 matrix.</p>"}, "variables": {"a11": {"name": "a11", "group": "Ungrouped variables", "templateType": "randrange", "description": "", "definition": "random(1..10#1)"}, "a21": {"name": "a21", "group": "Ungrouped variables", "templateType": "randrange", "description": "", "definition": "random(1..5#1)"}, "lambda1": {"name": "lambda1", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "min({c1},{a11}+{a22}-{c1})"}, "k": {"name": "k", "group": "Ungrouped variables", "templateType": "randrange", "description": "", "definition": "random(1..6#1)"}, "a22": {"name": "a22", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "{k}*{a21}+{c1}"}, "c1": {"name": "c1", "group": "Ungrouped variables", "templateType": "randrange", "description": "", "definition": "random(14..20#1)"}, "lambda2": {"name": "lambda2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "max({c1},{a11}+{a22}-{c1})"}, "a12": {"name": "a12", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "k*(a11-c1)"}}, "functions": {}, "extensions": [], "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>", "name": "Eigenvalues & eigenvectors", "rulesets": {}, "ungrouped_variables": ["a11", "c1", "a12", "k", "a21", "a22", "lambda1", "lambda2"], "tags": [], "type": "question", "contributors": [{"name": "Matthew Mears", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1688/"}]}]}], "contributors": [{"name": "Matthew Mears", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1688/"}]}