// Numbas version: finer_feedback_settings
{"name": "Determinant of 3x3 Matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Determinant of 3x3 Matrix", "tags": [], "metadata": {"description": "<p>Exercises on calculating the determinant of 3x3 matrices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h2>$3 \\times 3$ Matrix Determinants</h2>\n<p>Calculate the determinants of the following matrices.</p>", "advice": "<p>For simplicity, we will use the expansion of the first row to calculate our determinant</p>\n<p><strong>a)</strong></p>\n<p>We have:</p>\n<p>$$<br/>\\begin{aligned}<br/>\\det{A} &amp;= a_1A_1 + b_1B_1+ c_1C_1 \\\\<br/>&amp;= \\var{maA[0][0]} <br/>\\begin{vmatrix} <br/>\\var{maA[1][1]} &amp; \\var{maA[1][2]} \\\\ <br/>\\var{maA[2][1]} &amp; \\var{maA[2][2]} \\\\<br/>\\end{vmatrix} - \\var{maA[0][1]} \\begin{vmatrix}<br/>\\var{maA[1][0]} &amp; \\var{maA[1][2]} \\\\<br/>\\var{maA[2][0]} &amp; \\var{maA[2][2]} \\\\<br/>\\end{vmatrix} + \\var{maA[0][2]} \\begin{vmatrix}<br/>\\var{maA[1][0]} &amp; \\var{maA[1][1]} \\\\<br/>\\var{maA[2][0]} &amp; \\var{maA[2][1]} \\\\<br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{maA[0][0]} \\times \\var{a1a} - \\left(\\var{maA[0][1]} \\times \\var{b1a}\\right) + \\var{maA[0][2]} \\times \\var{c1a} \\\\<br/>&amp;= \\var{deta}<br/>\\end{aligned}<br/>$$</p>\n<p><strong>b)</strong></p>\n<p>We have:</p>\n<p>$$<br/>\\begin{aligned}<br/>\\det{B} &amp;= a_1A_1 + b_1B_1+ c_1C_1 \\\\<br/>&amp;= \\var{maB[0][0]} <br/>\\begin{vmatrix} <br/>\\var{maB[1][1]} &amp; \\var{maB[1][2]} \\\\ <br/>\\var{maB[2][1]} &amp; \\var{maB[2][2]} \\\\<br/>\\end{vmatrix} - \\var{maB[0][1]} \\begin{vmatrix}<br/>\\var{maB[1][0]} &amp; \\var{maB[1][2]} \\\\<br/>\\var{maB[2][0]} &amp; \\var{maB[2][2]} \\\\<br/>\\end{vmatrix} + \\var{maB[0][2]} \\begin{vmatrix}<br/>\\var{maB[1][0]} &amp; \\var{maB[1][1]} \\\\<br/>\\var{maB[2][0]} &amp; \\var{maB[2][1]} \\\\<br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{maB[0][0]} \\times \\var{a1b} - \\left(\\var{maB[0][1]} \\times \\var{b1b}\\right) + \\var{maB[0][2]} \\times \\var{c1b} \\\\<br/>&amp;= \\var{detb}<br/>\\end{aligned}<br/>$$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"maA": {"name": "maA", "group": "A", "definition": "matrixrandom(3,3)", "description": "", "templateType": "anything", "can_override": false}, "maB": {"name": "maB", "group": "B", "definition": "matrixrandom(3,3)", "description": "", "templateType": "anything", "can_override": false}, "A1a": {"name": "A1a", "group": "A", "definition": "maA[1][1]*maA[2][2]-maA[2][1]*maA[1][2]", "description": "", "templateType": "anything", "can_override": false}, "B1a": {"name": "B1a", "group": "A", "definition": "maA[1][0]*maA[2][2]-maA[2][0]*maA[1][2]", "description": "", "templateType": "anything", "can_override": false}, "C1a": {"name": "C1a", "group": "A", "definition": "maA[1][0]*maA[2][1]-maA[1][1]*maA[2][0]", "description": "", "templateType": "anything", "can_override": false}, "deta": {"name": "deta", "group": "A", "definition": "det(maa)", "description": "", "templateType": "anything", "can_override": false}, "a1b": {"name": "a1b", "group": "Ungrouped variables", "definition": "maB[1][1]*maB[2][2]-maB[2][1]*maB[1][2]", "description": "", "templateType": "anything", "can_override": false}, "b1b": {"name": "b1b", "group": "Ungrouped variables", "definition": "maB[1][0]*maB[2][2]-maB[2][0]*maB[1][2]", "description": "", "templateType": "anything", "can_override": false}, "c1b": {"name": "c1b", "group": "Ungrouped variables", "definition": "maB[1][0]*maB[2][1]-maB[1][1]*maB[2][0]", "description": "", "templateType": "anything", "can_override": false}, "detb": {"name": "detb", "group": "Ungrouped variables", "definition": "det(maB)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1b", "b1b", "c1b", "detb"], "variable_groups": [{"name": "A", "variables": ["maA", "A1a", "B1a", "C1a", "deta"]}, {"name": "B", "variables": ["maB"]}], "functions": {"matrixrandom": {"parameters": [["rows", "integer"], ["columns", "number"]], "type": "matrix", "language": "jme", "definition": "matrix(repeat(repeat(random(0..9)*(random(1,1,1,-1)),columns),rows))"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the determinant of $A = \\simplify{{maA}}$</p>", "minValue": "det(maA)", "maxValue": "det(maA)", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the determinant of $B = \\simplify{{maB}}$</p>", "minValue": "det(maB)", "maxValue": "det(maB)", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Mark Patterson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5064/"}, {"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}]}]}], "contributors": [{"name": "Mark Patterson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5064/"}, {"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}]}