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{"name": "William's copy of Question 2.1 MATH 6005 Assessment 1 Matrix Multiplication 2 (3x3 by 3x2 matrices)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "variable_groups": [], "variables": {"b31": {"definition": "random(-1..1)", "name": "b31", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "a21": {"definition": "random(-2..2)", "name": "a21", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "ab11": {"definition": "a11*b11+a12*b21+a13*b31", "name": "ab11", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "b11": {"definition": "random(-3,-1,0,3)", "name": "b11", "description": "", "templateType": "anything", "group": "Ungrouped 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(3x3 by 3x2 matrices)", "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>", "tags": [], "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "prompt": "<p>$AB = \\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\var{b31}&amp;\\var{b32}\\end{pmatrix} = $ [[0]]</p>", "gaps": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [ab11,ab12],\n    [ab21,ab22]\n])", "allowResize": false, "showFeedbackIcon": true, "marks": "1", "variableReplacements": [], "markPerCell": false, "scripts": {}, "allowFractions": false, "numRows": "2", "tolerance": 0, "numColumns": "2", "type": "matrix", "correctAnswerFractions": false}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "prompt": "<p>$BA = \\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\var{b31}&amp;\\var{b32}\\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}\\\\ \\end{pmatrix}=$ [[0]]</p>", "gaps": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [ba11,ba12,ba13],\n    [ba21,ba22,ba23],\n    [ba31,ba32,ba33]\n])", "allowResize": false, "showFeedbackIcon": true, "marks": "1", "variableReplacements": [], "markPerCell": false, "scripts": {}, "allowFractions": false, "numRows": "3", "tolerance": 0, "numColumns": "3", "type": "matrix", "correctAnswerFractions": false}]}], "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>Do the following matrix problems :<br/>Let <br/>\\[A=\\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}\\\\ \\end{pmatrix},\\;\\; B=\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\var{b31}&amp;\\var{b32}\\end{pmatrix},\\;\\; \\]<br/>Calculate the following products of these matrices:</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Multiplication of $2 \\times 2$ matrices.</p>"}, "functions": {}, "type": "question", "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "William Lee", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2797/"}]}]}], "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "William Lee", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2797/"}]}