// Numbas version: exam_results_page_options {"name": "Marisa's copy of 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": [{"extensions": [], "parts": [{"showFeedbackIcon": true, "type": "gapfill", "gaps": [{"showFeedbackIcon": true, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "type": "numberentry", "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "minValue": "det(matrixA)", "maxValue": "det(matrixA)", "correctAnswerFraction": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReduced": false}], "scripts": {}, "prompt": "

What is the determinant of A=$\\var{matrixA}$?

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[[0]]

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Calculate $|X|=$ [[0]]

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Hence, calculate $x=$  [[1]]

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Calculate $|Y|=$[[0]]

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Hence, calculate ${y=}$  [[1]]

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Calculate $|Z|=$[[0]]

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Hence, calculate $z=$  [[1]]

", "variableReplacements": [], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0}], "functions": {}, "variables": {"matrixA2": {"group": "Cramer determinants", "definition": "matrix([a11,c1,a13],[a21,c2,a23],[a31,c3,a33])", "templateType": "anything", "name": "matrixA2", "description": ""}, "a33": {"group": "Ungrouped variables", "definition": "random(0..10)", "templateType": "anything", "name": "a33", "description": ""}, "a22": {"group": "Ungrouped variables", "definition": "random(0..10 except(a21*a12/a11))", "templateType": "anything", "name": "a22", "description": ""}, "matrixA3": {"group": "Cramer determinants", "definition": "matrix([a11,a12,c1],[a21,a22,c2],[a31,a32,c3])", "templateType": "anything", "name": "matrixA3", "description": ""}, "matrixA1": {"group": "Cramer determinants", "definition": "matrix([c1,a12,a13],[c2,a22,a23],[c3,a32,a33])", "templateType": "anything", "name": "matrixA1", "description": ""}, "a31": {"group": "Ungrouped variables", "definition": "random(0..10)", "templateType": "anything", "name": "a31", "description": ""}, "a32": {"group": "Ungrouped variables", "definition": "random(0..10)", "templateType": "anything", "name": "a32", "description": ""}, "c3": {"group": "Ungrouped variables", "definition": "a31*x1+a32*x2+a33*x3", "templateType": "anything", "name": "c3", "description": ""}, "x2": {"group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "templateType": "anything", "name": "x2", "description": ""}, "c2": {"group": "Ungrouped variables", "definition": "a21*x1+a22*x2+a23*x3", "templateType": "anything", "name": "c2", "description": ""}, "a11": {"group": "Ungrouped variables", "definition": "random(0..10)", "templateType": "anything", "name": "a11", "description": ""}, "a23": {"group": "Ungrouped variables", "definition": "random(0..10)", "templateType": "anything", "name": "a23", "description": ""}, "matrixA": {"group": "Ungrouped variables", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "templateType": "anything", "name": "matrixA", "description": ""}, "a13": {"group": "Ungrouped variables", "definition": "random(0..10)", "templateType": "anything", "name": "a13", "description": ""}, "a12": {"group": "Ungrouped variables", "definition": "random(0..10)", "templateType": "anything", "name": "a12", "description": ""}, "x3": {"group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "templateType": "anything", "name": "x3", "description": ""}, "a21": {"group": "Ungrouped variables", "definition": "random(0..10)", "templateType": "anything", "name": "a21", "description": ""}, "x1": {"group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "templateType": "anything", "name": "x1", "description": ""}, "c1": {"group": "Ungrouped variables", "definition": "a11*x1+a12*x2+a13*x3", "templateType": "anything", "name": "c1", "description": ""}}, "advice": "

If \\[  A=\\left( \\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\\\a_{21} & a_{22} & a_{23}\\\\ a_{31} & a_{32} & a_{33}\\end{array} \\right),\\]

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\\[  C=\\left( \\begin{array}{ccc}
c_{1} \\\\ c_{2} \\\\c_{3} \\end{array} \\right),\\]

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Cramer's Rule : ${x_1}=\\frac{\\Delta_1}{\\Delta_0}$ ,  ${x_2}=\\frac{\\Delta_2}{\\Delta_0}$ , ${x_3}=\\frac{\\Delta_3}{\\Delta_0}$

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Where:\\[ \\Delta_0=\\left| \\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\\\a_{21} & a_{22} & a_{23}\\\\ a_{31} & a_{32} & a_{33}\\end{array} \\right|\\]

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\\[ \\Delta_1=\\left| \\begin{array}{ccc}
c_{1} & a_{12} & a_{13} \\\\c_{2} & a_{22} & a_{23}\\\\ c_{3} & a_{32} & a_{33}\\end{array} \\right|\\]

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\\[ \\Delta_2=\\left| \\begin{array}{ccc}
a_{11} & c_{1} & a_{13} \\\\a_{21} & c_{2} & a_{23}\\\\ a_{31} & c_{3} & a_{33}\\end{array} \\right|\\]

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\\[ \\Delta_3=\\left| \\begin{array}{ccc}
a_{11} & a_{12} & c_{1} \\\\a_{21} & a_{22} & c_{2}\\\\ a_{31} & a_{32} & c_{3}\\end{array} \\right|\\]

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", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Cramers Rule applied to 3 simultaneous equations

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Using Cramer's rule , solve the system of equations:

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$\\var{a11}x+\\var{a12}y+\\var{a13}z=\\var{c1}$

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$\\var{a21}x+\\var{a22}y+\\var{a23}z=\\var{c2}$

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$\\var{a31}x+\\var{a32}y+\\var{a33}z=\\var{c3}$

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