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{"name": "Determinant of 3 x 3 matrices", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"gaps": [{"notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 1, "scripts": {}, "allowFractions": false, "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(a)", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "det(a)", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": ""}], "showFeedbackIcon": true, "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "sortAnswers": false, "showCorrectAnswer": true, "prompt": "<p>$det(A) = $ [[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": ""}, {"gaps": [{"notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 1, "scripts": {}, "allowFractions": false, "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(b)", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "det(b)", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": ""}], "showFeedbackIcon": true, "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "sortAnswers": false, "showCorrectAnswer": true, "prompt": "<p>$det (B)&nbsp;=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": ""}, {"gaps": [{"notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 1, "scripts": {}, "allowFractions": false, "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(C)", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "det(C)", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": ""}], "showFeedbackIcon": true, "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "sortAnswers": false, "showCorrectAnswer": true, "prompt": "<p>$det(C)&nbsp;= $ [[0]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": ""}], "variable_groups": [], "tags": [], "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>", "advice": "<p>The determinant of a 3 x 3 matrix</p>\n<p>\\[A = \\begin{pmatrix} a_{11} \\ a_{12} \\ a_{13} \\\\ a_{21} \\ a_{22} \\ a_{23} \\\\ a_{31} \\ a_{32} \\ a_{33}&nbsp; &nbsp;\\end{pmatrix}\\]</p>\n<p>is given by</p>\n<p>\\[det(A)&nbsp; = a_{11}\\left| \\begin{matrix} a_{22} \\ a_{23} \\\\ a_{32} \\ a_{33}\\end{matrix}\\right| -&nbsp;a_{12}\\left| \\begin{matrix} a_{21} \\ a_{23} \\\\ a_{31} \\ a_{33}\\end{matrix}\\right| +&nbsp;a_{13}\\left| \\begin{matrix} a_{21} \\ a_{22} \\\\ a_{31} \\ a_{32}\\end{matrix}\\right| &nbsp;\\]</p>\n<p></p>\n<p>This is one way of finding the determinant of a matrix. We can choose any row or column, provided it corresponds with the sign matrix, to calculate the determinant.</p>\n<p></p>\n<p>\\[\\text{Sign matrix} = \\begin{pmatrix}+ \\ - \\ + \\\\ -\\ + \\ - \\\\ + \\ - \\ + \\end{pmatrix} \\]</p>\n<p></p>\n<p>For further information see Section 4 of the Chapter 10 Notes.&nbsp;</p>\n<p></p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noleadingminus"]}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Three examples of determinant of 2x2 matrices.</p>"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["a", "q1", "c", "b", "r1", "q", "p", "p1", "apb", "lcab", "lcabc", "p2", "q2", "apb1", "p3", "q3", "apb2", "lcabc2"], "preamble": {"css": "", "js": ""}, "functions": {}, "extensions": ["stats"], "variables": {"p2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..10 except p1)", "description": "", "name": "p2"}, "q3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6 except p)", "description": "", "name": "q3"}, "r1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6 except [0,1,-1,p1,q1])", "description": "", "name": "r1"}, "lcab": {"group": "Ungrouped variables", "templateType": "anything", "definition": "p*a+q*b", "description": "", "name": "lcab"}, "q1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6 except [0,1,-1,p1,q])", "description": "", "name": "q1"}, "apb": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a+b", "description": "", "name": "apb"}, "apb1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "p2*a + q2*b", "description": "<p>apb</p>", "name": "apb1"}, "lcabc2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "p1*a+q2*b+q1*c", "description": "<p>cabc</p>", "name": "lcabc2"}, "lcabc": {"group": "Ungrouped variables", "templateType": "anything", "definition": "p1*a+q1*b-r1*c", "description": "", "name": "lcabc"}, "p3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6 except p)", "description": "<p>p2</p>", "name": "p3"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix(repeat(repeat(random(-5..5),3),3))", "description": "", "name": "c"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix(repeat(repeat(random(-5..5 except 0) ,3),3))", "description": "", "name": "a"}, "q2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-6..6 except [0,1,-1,p])", "description": "<p>q</p>", "name": "q2"}, "apb2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "p3*a + q3*b", "description": "<p>apb</p>", "name": "apb2"}, "q": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6 except [0,1,-1,p])", "description": "", "name": "q"}, "p1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6 except p)", "description": "", "name": "p1"}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6)", "description": "", "name": "p"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix(repeat(repeat(random(-5..5 except 0),3),3))", "description": "", "name": "b"}}, "name": "Determinant of 3 x 3 matrices", "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/"}]}