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{"name": "Week 13: Matrices:  Cramers Rule 3x3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"a21": {"definition": "random(0..10)", "name": "a21", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a12": {"definition": "random(0..10)", "name": "a12", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a22": {"definition": "random(0..10 except(a21*a12/a11))", "name": "a22", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c3": {"definition": "a31*x1+a32*x2+a33*x3", "name": "c3", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "matrixA2": {"definition": "matrix([a11,c1,a13],[a21,c2,a23],[a31,c3,a33])", "name": "matrixA2", "group": "Cramer determinants", "templateType": "anything", "description": ""}, "x3": {"definition": "random(-10..10 except 0)", "name": "x3", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a32": {"definition": "random(0..10)", "name": "a32", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a23": {"definition": "random(0..10)", "name": "a23", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a11": {"definition": "random(0..10)", "name": "a11", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "x1": {"definition": "random(-10..10 except 0)", "name": "x1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a13": {"definition": "random(0..10)", "name": "a13", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "matrixA3": {"definition": "matrix([a11,a12,c1],[a21,a22,c2],[a31,a32,c3])", "name": "matrixA3", "group": "Cramer determinants", "templateType": "anything", "description": ""}, "x2": {"definition": "random(-10..10 except 0)", "name": "x2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "matrixA": {"definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "name": "matrixA", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c2": {"definition": "a21*x1+a22*x2+a23*x3", "name": "c2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a31": {"definition": "random(0..10)", "name": "a31", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "matrixA1": {"definition": "matrix([c1,a12,a13],[c2,a22,a23],[c3,a32,a33])", "name": "matrixA1", "group": "Cramer determinants", "templateType": "anything", "description": ""}, "a33": {"definition": "random(0..10)", "name": "a33", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c1": {"definition": "a11*x1+a12*x2+a13*x3", "name": "c1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "functions": {}, "tags": [], "ungrouped_variables": ["matrixA", "a11", "a12", "a21", "a22", "a13", "a23", "a31", "a32", "a33", "x1", "x2", "x3", "c1", "c2", "c3"], "parts": [{"prompt": "<p>What is the determinant of A=$\\var{matrixA}$?</p>\n<p>[[0]]</p>", "type": "gapfill", "variableReplacements": [], "gaps": [{"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "det(matrixA)", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false, "scripts": {}, "correctAnswerStyle": "plain", "minValue": "det(matrixA)", "marks": 1}], "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0}, {"prompt": "<p>Calculate&nbsp;$|X|=$ [[0]]</p>\n<p>Hence, calculate $x=$&nbsp; [[1]]</p>", "type": "gapfill", "variableReplacements": [], "gaps": [{"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "det(matrixA1)", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false, "scripts": {}, "correctAnswerStyle": "plain", "minValue": "det(matrixA1)", "marks": 1}, {"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{x1}", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false, "scripts": {}, "correctAnswerStyle": "plain", "minValue": "{x1}", "marks": 1}], "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0}, {"prompt": "<p>Calculate&nbsp;$|Y|=$[[0]]</p>\n<p>Hence, calculate ${y=}$&nbsp; [[1]]</p>", "type": "gapfill", "variableReplacements": [], "gaps": [{"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "det(matrixA2)", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false, "scripts": {}, "correctAnswerStyle": "plain", "minValue": "det(matrixA2)", "marks": 1}, {"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{x2}", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false, "scripts": {}, "correctAnswerStyle": "plain", "minValue": "{x2}", "marks": 1}], "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0}, {"prompt": "<p style=\"text-align: left;\">Calculate&nbsp;$|Z|=$[[0]]</p>\n<p>Hence, calculate $z=$&nbsp; [[1]]</p>", "type": "gapfill", "variableReplacements": [], "gaps": [{"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "det(matrixA3)", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false, "scripts": {}, "correctAnswerStyle": "plain", "minValue": "det(matrixA3)", "marks": 1}, {"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{x3}", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false, "scripts": {}, "correctAnswerStyle": "plain", "minValue": "{x3}", "marks": 1}], "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Week 13: Matrices:  Cramers Rule 3x3", "advice": "<p>If \\[&nbsp; A=\\left( \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; a_{13} \\\\a_{21} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right),\\]</p>\n<p>\\[&nbsp; C=\\left( \\begin{array}{ccc}<br/>c_{1} \\\\&nbsp;c_{2}&nbsp;\\\\c_{3} \\end{array} \\right),\\]</p>\n<p>Cramer's Rule : ${x_1}=\\frac{\\Delta_1}{\\Delta_0}$ ,&nbsp; ${x_2}=\\frac{\\Delta_2}{\\Delta_0}$ , ${x_3}=\\frac{\\Delta_3}{\\Delta_0}$</p>\n<p>Where:\\[ \\Delta_0=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; a_{13} \\\\a_{21} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_1=\\left| \\begin{array}{ccc}<br/>c_{1} &amp;&nbsp;a_{12} &amp; a_{13} \\\\c_{2} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ c_{3} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_2=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;c_{1} &amp; a_{13} \\\\a_{21} &amp;&nbsp;c_{2} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;c_{3} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_3=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; c_{1} \\\\a_{21} &amp;&nbsp;a_{22} &amp; c_{2}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; c_{3}\\end{array} \\right|\\]</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p><span class=\"nb\"><span class=\"k\"><span class=\"s\">&nbsp;</span></span></span></p>\n<p></p>", "extensions": [], "rulesets": {}, "preamble": {"js": "", "css": ""}, "variable_groups": [{"variables": ["matrixA1", "matrixA2", "matrixA3"], "name": "Cramer determinants"}], "statement": "<p>Using Cramer's rule , solve the system of equations:</p>\n<p>$\\var{a11}x+\\var{a12}y+\\var{a13}z=\\var{c1}$</p>\n<p>$\\var{a21}x+\\var{a22}y+\\var{a23}z=\\var{c2}$</p>\n<p>$\\var{a31}x+\\var{a32}y+\\var{a33}z=\\var{c3}$</p>\n<p></p>\n<p></p>", "metadata": {"description": "<p>Cramers Rule applied to 3 simultaneous equations</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "contributors": [{"name": "Ashley Cusack", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1818/"}]}]}], "contributors": [{"name": "Ashley Cusack", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1818/"}]}