// Numbas version: exam_results_page_options
{"name": "Ex 2 Matrix Multiplication 1 (2x2 matrices)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"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>", "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "<p>Multiplication of $2 \\times 2$ matrices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "statement": "<p>Do the following matrix problems <br/>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}\\]<br/>Calculate the following products of these matrices:</p>", "functions": {}, "variable_groups": [], "tags": [], "parts": [{"type": "gapfill", "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "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>", "gaps": [{"type": "matrix", "showFeedbackIcon": true, "numRows": "2", "correctAnswer": "matrix([\n    [ab11,ab12],\n    [ab21,ab22]\n])", "allowResize": false, "markPerCell": false, "showCorrectAnswer": true, "marks": 1, "allowFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "tolerance": 0, "numColumns": "2", "scripts": {}, "correctAnswerFractions": false}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true}, {"type": "gapfill", "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "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>", "gaps": [{"type": "matrix", "showFeedbackIcon": true, "numRows": "2", "correctAnswer": "matrix([\n    [ba11,ba12],\n    [ba21,ba22]\n])", "allowResize": false, "markPerCell": false, "showCorrectAnswer": true, "marks": 1, "allowFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "tolerance": 0, "numColumns": "2", "scripts": {}, "correctAnswerFractions": false}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true}, {"type": "gapfill", "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "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>", "gaps": [{"type": "matrix", "showFeedbackIcon": true, "numRows": "2", "correctAnswer": "matrix([\n    [cb11,cb12],\n    [cb21,cb22]\n])", "allowResize": false, "markPerCell": false, "showCorrectAnswer": true, "marks": 1, "allowFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "tolerance": 0, "numColumns": "2", "scripts": {}, "correctAnswerFractions": false}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true}, {"type": "gapfill", "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "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>", "gaps": [{"type": "matrix", "showFeedbackIcon": true, "numRows": "2", "correctAnswer": "matrix([\n    [ac11,ac12],\n    [ac21,ac22]\n])", "allowResize": false, "markPerCell": false, "showCorrectAnswer": true, "marks": 1, "allowFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "tolerance": 0, "numColumns": "2", "scripts": {}, "correctAnswerFractions": false}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true}], "name": "Ex 2 Matrix Multiplication 1 (2x2 matrices)", "extensions": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "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", 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