// Numbas version: exam_results_page_options
{"name": "Matrices: Basics (Instructional)", "metadata": {"description": "Basic definitions: Order, elements, Trace and Transpose.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Matrices: Basics 01", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "Classifying matrices (dimensions/order)", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<h4 style=\"text-align: left;\">The General Matrix</h4>\n<p style=\"text-align: left;\">A general $m \\times n$ matrix $A$ has $m$ rows and $n$ columns.<br/>The entries in the matrix $A$ are called the elements of $A$.<br/>In matrix $A$ the element in row $i$ and column $j$ is denoted by $a_{ij}$ .</p>", "advice": "<h4>We are presented with random matrices and asked to \"classify\" them.&nbsp;</h4>\n<p>This simply means \"give their dimensions\" - how many rows and columns do they have?</p>\n<p>In mathematical language you need to know that:</p>\n<p style=\"text-align: center;\">A general $m \\times n$ matrix $A$ has $m$ rows and $n$ columns.</p>\n<p>In simpler terms, the size is ALWAYS given as:</p>\n<p style=\"text-align: center;\">$\\Large ROWS \\times COLUMNS $</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Once you remember this, these are very straightforward.</p>\n<p style=\"text-align: left;\">&nbsp;&nbsp;</p>\n<p style=\"text-align: left;\">$A=\\var{A}$&nbsp; &nbsp; &nbsp;$A$ has $\\var{n1}$ rows and $\\var{m1}$ columns. So $A$ has dimensions $&nbsp;\\var{n1}&nbsp; \\times&nbsp; \\var{m1}$</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$B=\\var{B}$&nbsp; &nbsp; &nbsp;$B$ has $\\var{n2}$ rows and $\\var{m2}$ columns. So $B$ has dimensions $&nbsp;\\var{n2}&nbsp; \\times&nbsp;\\var{m2}$</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$C=\\var{C}$&nbsp; &nbsp; &nbsp;$C$ has $\\var{n3}$ rows and $\\var{m3}$ columns. So $C$ has dimensions $&nbsp;\\var{n3}&nbsp; \\times&nbsp;\\var{m3}$</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$D=\\var{D}$&nbsp; &nbsp; &nbsp;$D$ has $\\var{n4}$ rows and $\\var{m4}$ columns. So $D$ has dimensions $&nbsp;\\var{n4}&nbsp; \\times&nbsp;\\var{m4}$</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$E=\\var{EE}$&nbsp; &nbsp; &nbsp;$E$ has $\\var{n5}$ rows and $\\var{m5}$ columns. So $E$ has dimensions $&nbsp;\\var{n5}&nbsp; \\times&nbsp;\\var{m5}$</p>", "rulesets": {}, "variables": {"m1": {"name": "m1", "group": "A", "definition": "random(1..2)", "description": "", "templateType": "anything"}, "A": {"name": "A", "group": "A", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n1),m1)))", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "B", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "m2": {"name": "m2", "group": "B", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "B", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n2),m2)))", "description": "", "templateType": "anything"}, "n3": {"name": "n3", "group": "C", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "m3": {"name": "m3", "group": "C", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "C": {"name": "C", "group": "C", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n3),m3)))", "description": "", "templateType": "anything"}, "n4": {"name": "n4", "group": "D", "definition": "random(1..(n3-1) except n3)", "description": "", "templateType": "anything"}, "m4": {"name": "m4", "group": "D", "definition": "random(1..(m3-1) except m3)", "description": "", "templateType": "anything"}, "n1": {"name": "n1", "group": "A", "definition": "random(1..(m3-1) except m3)", "description": "", "templateType": "anything"}, "D": {"name": "D", "group": "D", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n4),m4)))", "description": "", "templateType": "anything"}, "n5": {"name": "n5", "group": "E", "definition": "m3", "description": "", "templateType": "anything"}, "m5": {"name": "m5", "group": "E", "definition": "n4", "description": "", "templateType": "anything"}, "EE": {"name": "EE", "group": "E", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n5),m5)))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "A", "variables": ["n1", "m1", "A"]}, {"name": "B", "variables": ["n2", "m2", "B"]}, {"name": "C", "variables": ["n3", "m3", "C"]}, {"name": "D", "variables": ["n4", "m4", "D"]}, {"name": "E", "variables": ["n5", "m5", "EE"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h4 style=\"text-align: left;\">Classify the following matrices:</h4>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: center;\">$A=\\var{A}$</p>\n<p style=\"text-align: center;\">$A$ is a&nbsp;[[0]]$\\times$ [[1]] matrix.</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\">$B=\\var{B}$</p>\n<p style=\"text-align: center;\">$B$ is a&nbsp;[[2]]$\\times$ [[3]] matrix.</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\">$C=\\var{C}$</p>\n<p style=\"text-align: center;\">$C$ is a&nbsp;[[4]]$\\times$ [[5]] matrix.</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\">$D=\\var{D}$</p>\n<p style=\"text-align: center;\">$D$ is a [[6]]$\\times$ [[7]] matrix.</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\">$E=\\var{EE}$</p>\n<p style=\"text-align: center;\">$E$ is a [[8]]$\\times$ [[9]] matrix.</p>\n<p style=\"text-align: center;\">&nbsp;</p>", "gaps": 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false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{m1}", "maxValue": "{m1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{n2}", "maxValue": "{n2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{m2}", "maxValue": "{m2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{n3}", "maxValue": "{n3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{m3}", "maxValue": "{m3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{n4}", "maxValue": "{n4}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{m4}", "maxValue": "{m4}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{n5}", "maxValue": "{n5}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{m5}", "maxValue": "{m5}", "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"}, {"name": "Matrices: Basics 02", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "Identify element of a matrix.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<h4 style=\"text-align: left;\">The General Matrix</h4>\n<p style=\"text-align: left;\">A general $m \\times n$ matrix $A$ has $m$ rows and $n$ columns.<br/>The entries in the matrix $A$ are called the elements of $A$.<br/>In matrix $A$ the element in row $i$ and column $j$ is denoted by $a_{ij}$ .</p>", "advice": "<h4>We are presented with random matrices and asked to identify certain elements of those matrices.</h4>\n<p>What you need to remember is:</p>\n<p style=\"text-align: center;\">In matrix $A$ the element in row $i$ and column $j$ is denoted by $a_{ij}$ .</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: left;\">In easier terms, in the subscript the numbers represent <strong>row then column</strong>.&nbsp;</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">So given the matrix&nbsp; $A=\\var{A}$</p>\n<p style=\"text-align: left;\">$a_{\\var{n1}\\var{m1}}$ is the element in row $\\var{n1}$ and column $\\var{m1}$. Therefore,&nbsp; $a_{\\var{n1}\\var{m1}}=\\var{E1}$</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">given the matrix&nbsp; $B=\\var{B}$</p>\n<p style=\"text-align: left;\">$b_{\\var{n2}\\var{m2}}$ is the element in row $\\var{n2}$ and column $\\var{m2}$. Therefore,&nbsp; $b_{\\var{n2}\\var{m2}}=\\var{E2}$</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">given the matrix&nbsp; $C=\\var{C}$</p>\n<p style=\"text-align: left;\">$c_{\\var{n3}\\var{m3}}$&nbsp;is the element in row $\\var{n3}$ and column $\\var{m3}$. Therefore,&nbsp;&nbsp;$b_{\\var{n2}\\var{m2}}=\\var{E3}$</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$c_{\\var{n4}\\var{m4}}$&nbsp;is the element in row $\\var{n4}$ and column $\\var{m4}$. Therefore,&nbsp;&nbsp;$b_{\\var{n2}\\var{m2}}=\\var{E4}$</p>\n<p></p>", "rulesets": {}, "variables": {"m1": {"name": "m1", "group": "A", "definition": "random(1..2)", "description": "", "templateType": "anything"}, "A": {"name": "A", "group": "A", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n1),m1)))", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "B", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "m2": {"name": "m2", "group": "B", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "B", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n2),m2)))", "description": "", "templateType": "anything"}, "n3": {"name": "n3", "group": "C", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "m3": {"name": "m3", "group": "C", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "C": {"name": "C", "group": "C", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n3),m3)))", "description": "", "templateType": "anything"}, "n4": {"name": "n4", "group": "D", "definition": "random(1..(n3-1) except n3)", "description": "", "templateType": "anything"}, "m4": {"name": "m4", "group": "D", "definition": "random(1..(m3-1) except m3)", "description": "", "templateType": "anything"}, "n1": {"name": "n1", "group": "A", "definition": "random(1..(m3-1) except m3)", "description": "", "templateType": "anything"}, "D": {"name": "D", "group": "D", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n4),m4)))", "description": "", "templateType": "anything"}, "n5": {"name": "n5", "group": "E", "definition": "m3", "description": "", "templateType": "anything"}, "m5": {"name": "m5", "group": "E", "definition": "n4", "description": "", "templateType": "anything"}, "EE": {"name": "EE", "group": "E", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n5),m5)))", "description": "", "templateType": "anything"}, "e1": {"name": "e1", "group": "A", "definition": "A[n1-1][m1-1]", "description": "", "templateType": "anything"}, "E2": {"name": "E2", "group": "B", "definition": "b[n2-1][m2-1]", "description": "", "templateType": "anything"}, "E3": {"name": "E3", "group": "C", "definition": "c[n3-1][m3-1]", "description": "", "templateType": "anything"}, "E4": {"name": "E4", "group": "C", "definition": "c[n4-1][m4-1]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "A", "variables": ["n1", "m1", "A", "e1"]}, {"name": "B", "variables": ["n2", "m2", "B", "E2"]}, {"name": "C", "variables": ["n3", "m3", "C", "E3", "E4"]}, {"name": "D", "variables": ["n4", "m4", "D"]}, {"name": "E", "variables": ["n5", "m5", "EE"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h4 style=\"text-align: left;\">Given the matrices:</h4>\n<p style=\"text-align: center;\">&nbsp;$A=\\var{A}$&nbsp; &nbsp; &nbsp;$B=\\var{B}$&nbsp; &nbsp; &nbsp;$C=\\var{C}$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">&nbsp;Give the values of the following elements of the matrices above:</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$a_{\\var{n1}\\var{m1}}=$ [[0]]</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$b_{\\var{n2}\\var{m2}}=$ [[1]]</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$c_{\\var{n3}\\var{m3}}=$ [[2]]</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$c_{\\var{n4}\\var{m4}}=$ [[3]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "A[n1-1][m1-1]", "maxValue": "A[n1-1][m1-1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b[n2-1][m2-1]", "maxValue": "b[n2-1][m2-1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c[n3-1][m3-1]", "maxValue": "c[n3-1][m3-1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c[n4-1][m4-1]", "maxValue": "c[n4-1][m4-1]", "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"}, {"name": "Matrices: Basics 03", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "Square matrices, leading/principle diagonal and trace.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<h4 style=\"text-align: left;\">Square Matrices</h4>\n<p style=\"text-align: left;\">When the number of rows is the same as the number of colums i.e. $m=n$, the matrix is said to be <strong>square</strong> and of <strong>order</strong> $n$ (or $m$ since they are the same).</p>\n<p style=\"text-align: left;\">In an $n \\times n$ square matrix $A$, the <strong>leading diagonal</strong> (or <strong>principal diagonal</strong>) is the top-left to bottom right collection of elements&nbsp; &nbsp;$a_{11}, a_{22}, a_{33}, . . . ,a_{nn}$.</p>\n<p style=\"text-align: left;\">The sum of the elements in the leading diagonal of $A$ is called the <strong>trace</strong> of the matrix and we write it as $tr(A)$.</p>\n<p style=\"text-align: center;\">If&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $ A= \\left( \\begin{array}{ccc} a_{11}&nbsp;&amp; a_{12} &amp;&nbsp;... &amp; a_{1n}\\\\ a_{21}&nbsp;&amp; a_{22} &amp;&nbsp;... &amp; a_{2n}\\\\ \\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots\\\\&nbsp;a_{n1}&nbsp;&amp; a_{n2} &amp;&nbsp;... &amp; a_{nn} \\end{array}&nbsp;\\right)&nbsp; &nbsp;$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;then&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$ tr(A)=&nbsp;a_{11}+a_{22}+...+a_{nn}$</p>", "advice": "<h4>We are presented with random matrices and asked to calculate the trace of each one.</h4>\n<p></p>\n<p>Remembering that, for a square matrix, the trace is&nbsp;the sum of the elements in the leading diagonal.&nbsp;</p>\n<p>Begin at the top left element and work down to the bottom right element, adding as you go.</p>\n<p>$A=\\var{A}$</p>\n<p style=\"text-align: center;\">$tr(A)=\\var{A[0][0]}+\\var{A[1][1]} +\\var{A[2][2]}=\\var{trA}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $</p>\n<p>&nbsp;</p>\n<p>$B=\\var{B}$</p>\n<p style=\"text-align: center;\">$tr(B)=\\var{B[0][0]}+\\var{B[1][1]}=\\var{trB}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>$C=\\var{C}$</p>\n<p style=\"text-align: center;\">$tr(C)=\\var{C[0][0]}+\\var{C[1][1]} +\\var{C[2][2]}+\\var{C[3][3]} +\\var{C[4][4]}=\\var{trC}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>$D=\\var{D}$</p>\n<p style=\"text-align: center;\">$tr(D)=\\var{D[0][0]}+\\var{D[1][1]} +\\var{D[2][2]}+\\var{D[3][3]}=\\var{trD}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>$E=\\var{EE}$</p>\n<p style=\"text-align: center;\">$tr(E)=\\var{EE[0][0]}+\\var{EE[1][1]} +\\var{EE[2][2]}=\\var{trE}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $</p>", "rulesets": {}, "variables": {"m1": {"name": "m1", "group": "A", "definition": "n1", "description": "", "templateType": "anything"}, "A": {"name": "A", "group": "A", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n1),m1)))", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "B", "definition": "2", "description": "", "templateType": "number"}, "m2": {"name": "m2", "group": "B", "definition": "n2", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "B", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n2),m2)))", "description": "", "templateType": "anything"}, "n3": {"name": "n3", "group": "C", "definition": "5", "description": "", "templateType": "number"}, "m3": {"name": "m3", "group": "C", "definition": "n3", "description": "", "templateType": "anything"}, "C": {"name": "C", "group": "C", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n3),m3)))", "description": "", "templateType": "anything"}, "n4": {"name": "n4", "group": "D", "definition": "m4", "description": "", "templateType": "anything"}, "m4": {"name": "m4", "group": "D", "definition": "4", "description": "", "templateType": "number"}, "n1": {"name": "n1", "group": "A", "definition": "3", "description": "", "templateType": "number"}, "D": {"name": "D", "group": "D", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n4),m4)))", "description": "", "templateType": "anything"}, "n5": {"name": "n5", "group": "E", "definition": "n1", "description": "", "templateType": "anything"}, "m5": {"name": "m5", "group": "E", "definition": "n1", "description": "", "templateType": "anything"}, "EE": {"name": "EE", "group": "E", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n5),m5)))", "description": "", "templateType": "anything"}, "e1": {"name": "e1", "group": "A", "definition": "A[n1-1][m1-1]", "description": "", "templateType": "anything"}, "E2": {"name": "E2", "group": "B", "definition": "b[n2-1][m2-1]", "description": "", "templateType": "anything"}, "E3": {"name": "E3", "group": "C", "definition": "c[n3-1][m3-1]", "description": "", "templateType": "anything"}, "E4": {"name": "E4", "group": "C", "definition": "c[n4-1][m4-1]", "description": "", "templateType": "anything"}, "trA": {"name": "trA", "group": "A", "definition": "(A[0][0])+(A[1][1])+(A[2][2])", "description": "", "templateType": "anything"}, "trB": {"name": "trB", "group": "B", "definition": "(B[0][0])+(B[1][1])", "description": "", "templateType": "anything"}, "trC": {"name": "trC", "group": "C", "definition": "(C[0][0])+(C[1][1])+(C[2][2])+(C[3][3])+(C[4][4])", "description": "", "templateType": "anything"}, "trD": {"name": "trD", "group": "D", "definition": "(D[0][0])+(D[1][1])+(D[2][2])+(D[3][3])", "description": "", "templateType": "anything"}, "trE": {"name": "trE", "group": "E", "definition": "(EE[0][0])+(EE[1][1])+(EE[2][2])", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "A", "variables": ["n1", "m1", "A", "e1", "trA"]}, {"name": "B", "variables": ["n2", "m2", "B", "E2", "trB"]}, {"name": "C", "variables": ["n3", "m3", "C", "E3", "E4", "trC"]}, {"name": "D", "variables": ["n4", "m4", "D", "trD"]}, {"name": "E", "variables": ["n5", "m5", "EE", "trE"]}], "functions": {"calculateTrace": {"parameters": [], "type": "number", "language": "javascript", "definition": ""}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h4>Calculate the trace of the following matrices:</h4>\n<p></p>\n<p>$A=\\var{A}$&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</p>\n<p style=\"text-align: center;\">$tr(A)=$ [[0]]</p>\n<p>&nbsp;</p>\n<p>$B=\\var{B}$&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</p>\n<p style=\"text-align: center;\">$tr(B)=$ [[1]]</p>\n<p>&nbsp;</p>\n<p>$C=\\var{C}$&nbsp; &nbsp; &nbsp; &nbsp;</p>\n<p style=\"text-align: center;\">$tr(C)=$ [[2]]</p>\n<p>&nbsp;</p>\n<p>$D=\\var{D}$&nbsp; &nbsp; &nbsp; &nbsp;</p>\n<p style=\"text-align: center;\">$tr(D)=$ [[3]]</p>\n<p>&nbsp;</p>\n<p>$E=\\var{EE}$&nbsp; &nbsp; &nbsp; &nbsp;</p>\n<p style=\"text-align: center;\">$tr(E)=$ [[4]]</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp;</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{trA}", "maxValue": "{trA}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{trB}", "maxValue": "{trB}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{trC}", "maxValue": "{trC}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{trD}", "maxValue": "{trD}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{trE}", "maxValue": "{trE}", "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"}, {"name": "Matrices: Basics 04", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Tran_Mat.gif", "/srv/numbas/media/question-resources/Tran_Mat.gif"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "Transpose of a matrix", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<h4 style=\"text-align: left;\">The transpose of a matrix $A$ is a matrix where the rows of $A$ become the columns of the new matrix&nbsp;and the columns of $A$ become its rows. For example:</h4>\n<p style=\"text-align: center;\">$&nbsp;A= \\left( \\begin{array}{ccc}&nbsp;1 &amp; 2 &amp; 3 \\\\&nbsp;4 &amp; 5 &amp; 6\\&nbsp;\\end{array}&nbsp;\\right)$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; becomes&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;$&nbsp;A^T= \\left(\\begin{array}{ccc}&nbsp;1 &amp; 4 \\\\ 2 &amp; 5 \\\\ 3&amp;6\\ \\end{array}&nbsp;\\right)$</p>\n<p style=\"text-align: left;\">The resulting matrix is called the <strong>transposed matrix</strong> of $A$ and is denoted $A^T$.</p>", "advice": "<h4>We are&nbsp;asked to work out the transpose of various matrices.</h4>\n<p>The transpose process results in rows becoming columns and columns becomimng rows.</p>\n<p>It may help to imagine the matrix being \"filpped\" about its diagonal.</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/Tran_Mat.gif\"/></p>\n<p style=\"text-align: left;\">$A=\\var{A}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $A^{T}=\\var{TA}$</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p>$B=\\var{B}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $B^{T}=\\var{TB}$</p>\n<p>&nbsp;</p>\n<p></p>\n<p>$C=\\var{C}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $C^{T}=\\var{TC}$</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p>$D=\\var{D}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $D^{T}=\\var{TD}$</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>$E=\\var{EE}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$E^{T}=\\var{TE}$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"m1": {"name": "m1", "group": "A", "definition": "random(1..2)", "description": "", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "A", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n1),m1)))", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "B", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": 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