// Numbas version: exam_results_page_options
{"name": "Matrices: Determinants 02", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Matrices: Determinants 02", "tags": [], "metadata": {"description": "<p>Determinant of 2x2 and notation</p>\n<p>Input answer only.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<h4>The determinant of the matrix:</h4>\n<p style=\"text-align: center;\">$\\large A= \\left( \\begin{array}{ccc} a&nbsp;&amp; b&nbsp; \\\\ c&nbsp;&amp; d&nbsp;&nbsp;\\end{array}&nbsp;\\right) $&nbsp; is denoted by&nbsp; &nbsp;$\\large \\left| \\begin{array}{ccc} a&nbsp;&amp; b&nbsp; \\\\ c&nbsp;&amp; d&nbsp;&nbsp;\\end{array}&nbsp;\\right| $</p>\n<p style=\"text-align: left;\">and is defined to be the number&nbsp; $ad-bc$. That is:</p>\n<p style=\"text-align: center;\">$\\large \\left| \\begin{array}{ccc} a&nbsp;&amp; b&nbsp; \\\\ c&nbsp;&amp; d&nbsp;&nbsp;\\end{array}&nbsp;\\right| =ad-bc$</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: left;\">We can use the notation&nbsp; $det(A)$&nbsp; or&nbsp; $|A|$&nbsp; or&nbsp; $\\Delta$&nbsp;&nbsp;to denote the determinant of $A$.</p>", "advice": "<h4>We are asked to calculate the determinants of some&nbsp; $2 \\times 2$&nbsp; matrices:</h4>\n<p>&nbsp;We simply have to carry out this calculation for each one,</p>\n<p style=\"text-align: center;\">$\\large \\left| \\begin{array}{ccc} a&nbsp;&amp; b&nbsp; \\\\ c&nbsp;&amp; d&nbsp;&nbsp;\\end{array}&nbsp;\\right| =ad-bc$</p>\n<p>&nbsp;$A=\\var{A}$</p>\n<p style=\"text-align: center;\">$|A|=\\left|\\begin{array}{ccc}\\var{A[0][0]} &amp; \\var{A[0][1]} \\\\\\var{A[1][0]} &amp; \\var{A[1][1]} \\end{array}\\right| =\\var{Apart1}\\large -\\var{Apart2}=\\var{detA}$</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>&nbsp;$B=\\var{B}$</p>\n<p style=\"text-align: center;\">$|B|=\\left|\\begin{array}{ccc}\\var{B[0][0]} &amp; \\var{B[0][1]} \\\\\\var{B[1][0]} &amp; \\var{B[1][1]} \\end{array}\\right| =\\var{Bpart1}\\large -\\var{Bpart2}=\\var{detB}$</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>&nbsp;$C=\\var{C}$</p>\n<p style=\"text-align: center;\">$|C|= \\left|\\begin{array}{ccc}\\var{C[0][0]} &amp; \\var{C[0][1]} \\\\\\var{C[1][0]} &amp; \\var{C[1][1]} \\end{array}\\right| =\\var{Cpart1}\\large -\\var{Cpart2}=\\var{detC}$</p>\n<p>&nbsp;</p>\n<p>$D=\\var{D}$</p>\n<p style=\"text-align: center;\">$|D|= \\left|\\begin{array}{ccc}\\var{D[0][0]} &amp; \\var{D[0][1]} \\\\\\var{D[1][0]} &amp; \\var{D[1][1]} \\end{array}\\right| =\\var{Dpart1}\\large -\\var{Dpart2}=\\var{detD}$</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\">&nbsp;</p>", "rulesets": {}, "extensions": [], "variables": {"n1": {"name": "n1", "group": "Ungrouped variables", "definition": "2", "description": "", "templateType": "number"}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "n1", "description": "", "templateType": "anything"}, "A": {"name": "A", "group": "Matrix A", "definition": "transpose(matrix(repeat(repeat(random(1..9),n1),m1)))", "description": "", "templateType": "anything"}, "detA": {"name": "detA", "group": "Matrix A", "definition": "det(A)", "description": "", "templateType": "anything"}, "Apart1": {"name": "Apart1", "group": "Matrix A", "definition": "(A[0][0])*(A[1][1])", "description": "", "templateType": "anything"}, "Apart2": {"name": "Apart2", "group": "Matrix A", "definition": "(A[0][1])*(A[1][0])", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),n1),m1)))", "description": "", "templateType": "anything"}, "Bpart1": {"name": "Bpart1", "group": "Matrix B", "definition": "(B[0][0])*(B[1][1])", "description": "", "templateType": "anything"}, "Bpart2": {"name": "Bpart2", "group": "Matrix B", "definition": "(B[0][1])*(B[1][0])", "description": "", "templateType": "anything"}, "detB": {"name": "detB", "group": "Matrix B", "definition": "det(B)", "description": "", "templateType": "anything"}, "C": {"name": "C", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n1),m1)))", "description": "", "templateType": "anything"}, "detC": {"name": "detC", "group": "Matrix C", "definition": "det(C)", "description": "", "templateType": "anything"}, "Cpart1": {"name": "Cpart1", "group": "Matrix C", "definition": "(C[0][0])*(C[1][1])", "description": "", "templateType": "anything"}, "Cpart2": {"name": "Cpart2", "group": "Matrix C", "definition": "(C[0][1])*(C[1][0])", "description": "", "templateType": "anything"}, "D": {"name": "D", "group": "Matrix ", "definition": "transpose(matrix(repeat(repeat(random(-9..-1),n1),m1)))", "description": "", "templateType": "anything"}, "Dpart1": {"name": "Dpart1", "group": "Matrix ", "definition": "(D[0][0])*(D[1][1])", "description": "", "templateType": "anything"}, "Dpart2": {"name": "Dpart2", "group": "Matrix ", "definition": "(D[0][1])*(D[1][0])", "description": "", "templateType": "anything"}, "detD": {"name": "detD", "group": "Matrix ", "definition": "det(D)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n1", "m1"], "variable_groups": [{"name": "Matrix A", "variables": ["A", "Apart1", "Apart2", "detA"]}, {"name": "Matrix B", "variables": ["B", "Bpart1", "Bpart2", "detB"]}, {"name": "Matrix C", "variables": ["C", "Cpart1", "Cpart2", "detC"]}, {"name": "Matrix ", "variables": ["D", "Dpart1", "Dpart2", "detD"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h4>Calculate the determinants of the following matrices:</h4>\n<p>&nbsp;</p>\n<p>&nbsp;$A=\\var{A}$</p>\n<p style=\"text-align: center;\">$|A|=\\left|\\begin{array}{ccc}\\var{A[0][0]} &amp; \\var{A[0][1]} \\\\\\var{A[1][0]} &amp; \\var{A[1][1]} \\end{array}\\right| =$ [[0]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>&nbsp;$B=\\var{B}$</p>\n<p style=\"text-align: center;\">$|B|=\\left|\\begin{array}{ccc}\\var{B[0][0]} &amp; \\var{B[0][1]} \\\\\\var{B[1][0]} &amp; \\var{B[1][1]} \\end{array}\\right| =$&nbsp;[[1]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>$C=\\var{C}$</p>\n<p style=\"text-align: center;\">$|C|=\\left|\\begin{array}{ccc}\\var{C[0][0]} &amp; \\var{C[0][1]} \\\\\\var{C[1][0]} &amp; \\var{C[1][1]} \\end{array}\\right|=$&nbsp;[[2]]</p>\n<p>&nbsp;</p>\n<p>$D=\\var{D}$</p>\n<p style=\"text-align: center;\">$|D|= \\left|\\begin{array}{ccc}\\var{D[0][0]} &amp; \\var{D[0][1]} \\\\\\var{D[1][0]} &amp; \\var{D[1][1]} \\end{array}\\right| =$&nbsp;[[3]]</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "gaps": [{"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, "minValue": "{detA}", "maxValue": "{detA}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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, "minValue": "{detB}", "maxValue": "{detB}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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, "minValue": "{detC}", "maxValue": "{detC}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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, "minValue": "{detD}", "maxValue": "{detD}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}