// Numbas version: exam_results_page_options
{"name": "Joseph's copy of Matrix Multiplication 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"c12": {"templateType": "anything", "name": "c12", "description": "", "definition": "a12+b12", "group": "Ungrouped variables"}, "b12": {"templateType": "anything", "name": "b12", "description": "", "definition": "random(-3..1)", "group": "Ungrouped variables"}, "a11": {"templateType": "anything", "name": "a11", "description": "", "definition": "random(-2,1,2)", "group": "Ungrouped variables"}, "c22": {"templateType": "anything", "name": "c22", "description": "", "definition": "random(0,1)", "group": "Ungrouped variables"}, "ba12": {"templateType": "anything", "name": "ba12", "description": "", "definition": "b11*a12+b12*a22", "group": "Ungrouped variables"}, "ab21": {"templateType": "anything", "name": "ab21", "description": "", "definition": "a21*b11+a22*b21", "group": "Ungrouped variables"}, "ac22": {"templateType": "anything", "name": "ac22", "description": "", "definition": "a21*c12+a22*c22", "group": "Ungrouped variables"}, "ba21": {"templateType": "anything", "name": "ba21", "description": "", "definition": "b21*a11+b22*a21", "group": "Ungrouped variables"}, "ac21": {"templateType": "anything", "name": "ac21", "description": "", "definition": "a21*c11+a22*c21", "group": "Ungrouped variables"}, "cb12": {"templateType": "anything", "name": "cb12", "description": "", "definition": "c11*b12+c12*b22", "group": "Ungrouped variables"}, "ba22": {"templateType": "anything", "name": "ba22", "description": "", "definition": "b21*a12+b22*a22", "group": "Ungrouped variables"}, "b21": {"templateType": "anything", "name": "b21", "description": "", "definition": "random(2,3)", "group": "Ungrouped variables"}, "ac11": {"templateType": "anything", "name": "ac11", "description": "", "definition": "a11*c11+a12*c21", "group": "Ungrouped variables"}, "cb21": {"templateType": "anything", "name": "cb21", "description": "", "definition": "c21*b11+c22*b21", "group": "Ungrouped variables"}, "c21": {"templateType": "anything", "name": "c21", "description": "", "definition": "random(2..5)", "group": "Ungrouped variables"}, "ab11": {"templateType": "anything", "name": "ab11", "description": "", "definition": "a11*b11+a12*b21", "group": "Ungrouped variables"}, "cb22": {"templateType": "anything", "name": "cb22", "description": "", "definition": "c21*b12+c22*b22", "group": "Ungrouped variables"}, "ab12": {"templateType": "anything", "name": "ab12", "description": "", "definition": "a11*b12+a12*b22", "group": "Ungrouped variables"}, "ba11": {"templateType": "anything", "name": "ba11", "description": "", "definition": "b11*a11+b12*a21", "group": "Ungrouped variables"}, "b22": {"templateType": "anything", "name": "b22", "description": "", "definition": "random(-3..-1)", "group": "Ungrouped variables"}, "a12": {"templateType": "anything", "name": "a12", "description": "", "definition": "random(1..4)", "group": "Ungrouped variables"}, "ac12": {"templateType": "anything", "name": "ac12", "description": "", "definition": "a11*c12+a12*c22", "group": "Ungrouped variables"}, "cb11": {"templateType": "anything", "name": "cb11", "description": "", "definition": "c11*b11+c12*b21", "group": "Ungrouped variables"}, "b11": {"templateType": "anything", "name": "b11", "description": "", "definition": "random(-3,-1,0,3)", "group": "Ungrouped variables"}, "a22": {"templateType": "anything", "name": "a22", "description": "", "definition": "random(1..3)", "group": "Ungrouped variables"}, "a21": {"templateType": "anything", "name": "a21", "description": "", "definition": "random(-2..2)", "group": "Ungrouped variables"}, "ab22": {"templateType": "anything", "name": "ab22", "description": "", "definition": "a21*b12+a22*b22", "group": "Ungrouped variables"}, "c11": {"templateType": "anything", "name": "c11", "description": "", "definition": "random(1,0,4)", "group": "Ungrouped variables"}}, "preamble": {"js": "", "css": ""}, "functions": {}, "statement": "<p>Let <br/>\\[A=\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix},\\;\\; B=\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix},\\;\\; C=\\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}\\]</p>\n<p><br/>Calculate the following products of these matrices:</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "extensions": [], "tags": [], "metadata": {"description": "<p>Multiplication of $2 \\times 2$ matrices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "advice": "<h4>a)</h4>\n<p>\\[ \\begin{eqnarray*} AB &amp;=&amp; \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{a11}{b11}+{a12}{b21}}&amp;\\simplify[]{{a11}{b12}+{a12}{b22}}\\\\ \\simplify[]{{a21}{b11}+{a22}{b21}}&amp;\\simplify[]{{a21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ab11}&amp;\\var{ab12}\\\\ \\var{ab21}&amp;\\var{ab22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>b)</h4>\n<p>\\[ \\begin{eqnarray*} BA &amp;=&amp; \\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{b11}{a11}+{b12}{a21}}&amp;\\simplify[]{{b11}{a12}+{b12}{a22}}\\\\ \\simplify[]{{b21}{a11}+{b22}{a21}}&amp;\\simplify[]{{b21}{a12}+{b22}{a22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ba11}&amp;\\var{ba12}\\\\ \\var{ba21}&amp;\\var{ba22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>c)</h4>\n<p>\\[ \\begin{eqnarray*} CB &amp;=&amp; \\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{c11}{b11}+{c12}{b21}}&amp;\\simplify[]{{c11}{b12}+{c12}{b22}}\\\\ \\simplify[]{{c21}{b11}+{c22}{b21}}&amp;\\simplify[]{{c21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{cb11}&amp;\\var{cb12}\\\\ \\var{cb21}&amp;\\var{cb22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>d)</h4>\n<p>\\[ \\begin{eqnarray*} AC &amp;=&amp; \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{a11}{c11}+{a12}{c21}}&amp;\\simplify[]{{a11}{c12}+{a12}{c22}}\\\\ \\simplify[]{{a21}{c11}+{a22}{c21}}&amp;\\simplify[]{{a21}{c12}+{a22}{c22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ac11}&amp;\\var{ac12}\\\\ \\var{ac21}&amp;\\var{ac22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>", "ungrouped_variables": ["ba21", "a21", "a22", "ba22", "cb21", "b22", "b21", "cb22", "ac22", "ac21", "ab22", "ab21", "b12", "b11", "c12", "c11", "c22", "a11", "cb11", "cb12", "a12", "c21", "ba11", "ba12", "ab12", "ab11", "ac12", "ac11"], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Joseph's copy of Matrix Multiplication 1", "parts": [{"type": "gapfill", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>$AB = \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix} = $ [[0]]</p>", "marks": 0, "scripts": {}, "variableReplacements": [], "gaps": [{"type": "matrix", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "numColumns": "2", "correctAnswerFractions": false, "customMarkingAlgorithm": "", "allowResize": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "numRows": "2", "marks": 1, "markPerCell": false, "scripts": {}, "tolerance": 0, "variableReplacements": [], "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [ab11,ab12],\n    [ab21,ab22]\n])"}]}, {"type": "gapfill", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>$BA = \\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}=$ [[0]]</p>", "marks": 0, "scripts": {}, "variableReplacements": [], "gaps": [{"type": "matrix", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "numColumns": "2", "correctAnswerFractions": false, "customMarkingAlgorithm": "", "allowResize": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "numRows": "2", "marks": 1, "markPerCell": false, "scripts": {}, "tolerance": 0, "variableReplacements": [], "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [ba11,ba12],\n    [ba21,ba22]\n])"}]}, {"type": "gapfill", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>$CB = \\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix} \\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}=$&nbsp;[[0]]</p>", "marks": 0, "scripts": {}, "variableReplacements": [], "gaps": [{"type": "matrix", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "numColumns": "2", "correctAnswerFractions": false, "customMarkingAlgorithm": "", "allowResize": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "numRows": "2", "marks": 1, "markPerCell": false, "scripts": {}, "tolerance": 0, "variableReplacements": [], "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [cb11,cb12],\n    [cb21,cb22]\n])"}]}, {"type": "gapfill", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>$AC = \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}=$&nbsp;[[0]]</p>", "marks": 0, "scripts": {}, "variableReplacements": [], "gaps": [{"type": "matrix", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "numColumns": "2", "correctAnswerFractions": false, "customMarkingAlgorithm": "", "allowResize": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "numRows": "2", "marks": 1, "markPerCell": false, "scripts": {}, "tolerance": 0, "variableReplacements": [], "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [ac11,ac12],\n    [ac21,ac22]\n])"}]}], "type": "question", "contributors": [{"name": "Joseph Clarke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2455/"}]}]}], "contributors": [{"name": "Joseph Clarke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2455/"}]}