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{"name": "Determinant of 2 x 2 matrices", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"q1": {"templateType": "anything", "name": "q1", "description": "", "definition": "random(1..6 except [0,1,-1,p1,q])", "group": "Ungrouped variables"}, "b": {"templateType": "anything", "name": "b", "description": "", "definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0]]),2),2))", "group": "Ungrouped variables"}, "q3": {"templateType": "anything", "name": "q3", "description": "", "definition": "random(2..6 except p)", "group": "Ungrouped variables"}, "lcab": {"templateType": "anything", "name": "lcab", "description": "", "definition": "p*a+q*b", "group": "Ungrouped variables"}, "a": {"templateType": "anything", "name": "a", "description": "", "definition": "matrix(repeat(repeat(random(-5..5 except 0),2),2))", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "name": "c", "description": "", "definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0],b[0][0]]),2),2))", "group": "Ungrouped variables"}, "p1": {"templateType": "anything", "name": "p1", "description": "", "definition": "random(2..6 except p)", "group": "Ungrouped variables"}, "p": {"templateType": "anything", "name": "p", "description": "", "definition": "random(2..6)", "group": "Ungrouped variables"}, "r1": {"templateType": "anything", "name": "r1", "description": "", "definition": "random(1..6 except [0,1,-1,p1,q1])", "group": "Ungrouped variables"}, "lcabc2": {"templateType": "anything", "name": "lcabc2", "description": "<p>cabc</p>", "definition": "p1*a+q2*b+q1*c", "group": "Ungrouped variables"}, "q": {"templateType": "anything", "name": "q", "description": "", "definition": "random(1..6 except [0,1,-1,p])", "group": "Ungrouped variables"}, "apb2": {"templateType": "anything", "name": "apb2", "description": "<p>apb</p>", "definition": "p3*a + q3*b", "group": "Ungrouped variables"}, "lcabc": {"templateType": "anything", "name": "lcabc", "description": "", "definition": "p1*a+q1*b-r1*c", "group": "Ungrouped variables"}, "q2": {"templateType": "anything", "name": "q2", "description": "<p>q</p>", "definition": "random(-6..6 except [0,1,-1,p])", "group": "Ungrouped variables"}, "p3": {"templateType": "anything", "name": "p3", "description": "<p>p2</p>", "definition": "random(2..6 except p)", "group": "Ungrouped variables"}, "p2": {"templateType": "anything", "name": "p2", "description": "", "definition": "random(2..10 except p1)", "group": "Ungrouped variables"}, "apb": {"templateType": "anything", "name": "apb", "description": "", "definition": "a+b", "group": "Ungrouped variables"}, "apb1": {"templateType": "anything", "name": "apb1", "description": "<p>apb</p>", "definition": "p2*a + q2*b", "group": "Ungrouped variables"}}, "preamble": {"js": "", "css": ""}, "functions": {}, "ungrouped_variables": ["a", "q1", "c", "b", "r1", "q", "p", "p1", "apb", "lcab", "lcabc", "p2", "q2", "apb1", "p3", "q3", "apb2", "lcabc2"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noleadingminus"]}, "extensions": ["stats"], "tags": [], "metadata": {"description": "<p>Three examples of determinant of 2x2 matrices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "advice": "<h4></h4>\n<p>The determinant of a 2 x 2 matrix</p>\n<p>\\[A = \\begin{pmatrix} a_{11} \\ a_{12} \\\\ a_{21} \\ a_{22}&nbsp; &nbsp; \\end{pmatrix}\\]</p>\n<p>is given by</p>\n<p>\\[det(A)&nbsp; = a_{11}a_{22} - a_{21}a_{12} \\]</p>\n<p>For each of the cases above:</p>\n<p>a)&nbsp;</p>\n<p>\\[det(A) = (\\var{a[0][0]}\\times\\var{a[1][1]}) - (\\var{a[1][0]}\\times\\var{a[0][1]}) = \\simplify{{a[0][0]}*{a[1][1]} - {a[1][0]}*{a[0][1]}}&nbsp; \\]</p>\n<p></p>\n<p>b)</p>\n<p>\\[det(B) = (\\var{b[0][0]}\\times\\var{b[1][1]}) - (\\var{b[1][0]}\\times\\var{b[0][1]}) = \\simplify{{b[0][0]}*{b[1][1]} - {b[1][0]}*{b[0][1]}}&nbsp; \\]</p>\n<p>c)</p>\n<p>\\[det(C) = (\\var{c[0][0]}\\times\\var{c[1][1]}) - (\\var{c[1][0]}\\times\\var{c[0][1]}) = \\simplify{{c[0][0]}*{c[1][1]} - {c[1][0]}*{c[0][1]}}&nbsp; \\]</p>\n<p>For further information see Section 4 of the Chapter 10 Notes.&nbsp;</p>", "statement": "<p>Let&nbsp;<br/>\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}},\\;\\; C=\\simplify{{c}}\\]<br/>Calculate the determinants of these matrices</p>\n<p>&nbsp;</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Determinant of 2 x 2 matrices", "parts": [{"type": "gapfill", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "unitTests": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>$det(A) = $ [[0]]</p>", "marks": 0, "scripts": {}, "variableReplacements": [], "gaps": [{"type": "numberentry", "variableReplacementStrategy": "originalfirst", "maxValue": "det(a)", "unitTests": [], "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "mustBeReducedPC": 0, "variableReplacements": [], "minValue": "det(a)"}]}, {"type": "gapfill", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "unitTests": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>$det (B)&nbsp;=$&nbsp;[[0]]</p>", "marks": 0, "scripts": {}, "variableReplacements": [], "gaps": [{"type": "numberentry", "variableReplacementStrategy": "originalfirst", "maxValue": "det(b)", "unitTests": [], "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "mustBeReducedPC": 0, "variableReplacements": [], "minValue": "det(b)"}]}, {"type": "gapfill", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "unitTests": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>$det(C)&nbsp;= $ [[0]]</p>\n<p></p>", "marks": 0, "scripts": {}, "variableReplacements": [], "gaps": [{"type": "numberentry", "variableReplacementStrategy": "originalfirst", "maxValue": "det(C)", "unitTests": [], "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "mustBeReducedPC": 0, "variableReplacements": [], "minValue": "det(C)"}]}], "type": "question", "contributors": [{"name": "Gemma Crowe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2440/"}, {"name": "Joseph Clarke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2455/"}]}]}], "contributors": [{"name": "Gemma Crowe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2440/"}, {"name": "Joseph Clarke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2455/"}]}