// Numbas version: exam_results_page_options
{"name": "Linear combinations of 2 x 2 matrices", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noleadingminus"]}, "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "statement": "<p>Let&nbsp;<br/>\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}},\\;\\; C=\\simplify{{c}}\\]<br/>Calculate the following $2 \\times 2$ matrices:</p>\n<p>&nbsp;</p>", "parts": [{"prompt": "<p>$\\mathrm{A}+\\mathrm{B} = \\simplify[std]{{a}+{b}} = $ [[0]]</p>", "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "gaps": [{"correctAnswerFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "numRows": "2", "tolerance": 0, "type": "matrix", "numColumns": "2", "allowFractions": false, "allowResize": false, "correctAnswer": "apb", "marks": "0.2", "markPerCell": false, "scripts": {}}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}, {"prompt": "<p>$\\simplify{{p1}A+{q1}B+{r1}C = {p1}{a}+{q1}{b}+{r1}{c}}=$ [[0]]</p>", "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "gaps": [{"correctAnswerFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "numRows": "2", "tolerance": 0, "type": "matrix", "numColumns": "2", "allowFractions": false, "allowResize": false, "correctAnswer": "lcabc", "marks": 1, "markPerCell": false, "scripts": {}}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}], "metadata": {"description": "<p>Linear combinations of $2 \\times 2$ matrices. Three examples.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variables": {"apb": {"description": "", "name": "apb", "group": "Ungrouped variables", "templateType": "anything", "definition": "a+b"}, "lcab": {"description": "", "name": "lcab", "group": "Ungrouped variables", "templateType": "anything", "definition": "p*a+q*b"}, "q": {"description": "", "name": "q", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-6..6 except [0,1,-1,p])"}, "b": {"description": "", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0]]),2),2))"}, "q1": {"description": "", "name": "q1", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-6..6 except [0,1,-1,p1,q])"}, "r1": {"description": "", "name": "r1", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-6..6 except [0,1,-1,p1,q1])"}, "lcabc": {"description": "", "name": "lcabc", "group": "Ungrouped variables", "templateType": "anything", "definition": "p1*a+q1*b+r1*c"}, "c": {"description": "", "name": "c", "group": "Ungrouped variables", "templateType": "anything", "definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0],b[0][0]]),2),2))"}, "p1": {"description": "", "name": "p1", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6 except p)"}, "a": {"description": "", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "definition": "matrix(repeat(repeat(random(-5..5 except 0),2),2))"}, "p": {"description": "", "name": "p", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6)"}}, "name": "Linear combinations of 2 x 2 matrices", "advice": "<h4>a)</h4>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{A+B} &amp;=&amp;\\simplify[std]{{a}+{b}}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{a[0][0]}+{b[0][0]}}&amp;&nbsp;\\simplify[std]{{a[0][1]}+{b[0][1]}}\\\\&nbsp;&nbsp;\\simplify[std]{{a[1][0]}+{b[1][0]}}&amp;\\simplify[std]{{a[1][1]}+{b[1][1]}} \\end{pmatrix}\\\\ &amp;=&amp;\\simplify{{apb}}\\\\ &nbsp;\\end{eqnarray*} \\]</p>\n<h4>b)&nbsp;</h4>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{{p}A+{q}B} &amp;=&amp;\\simplify[std]{{p}{a}+{q}{b}}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{p}*{a[0][0]}+{q}*{b[0][0]}}&amp;&nbsp;\\simplify[std]{{p}*{a[0][1]}+{q}*{b[0][1]}}\\\\&nbsp;&nbsp;\\simplify[std]{{p}*{a[1][0]}+{q}*{b[1][0]}}&amp;\\simplify[std]{{p}*{a[1][1]}+{q}*{b[1][1]}} \\end{pmatrix}\\\\ &amp;=&amp;\\simplify{{lcab}}\\\\ &nbsp;\\end{eqnarray*} \\]</p>\n<h4>c)</h4>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{{p1}A+{q1}B+{r1}C} &amp;=&amp;\\simplify[std]{{p1}{a}+{q1}{b}+{r1}{c}}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{p1}*{a[0][0]}+{q1}*{b[0][0]}+{r1}*{c[0][0]}}&amp;&nbsp;\\simplify[std]{{p1}*{a[0][1]}+{q1}*{b[0][1]}+{r1}*{c[0][1]}}\\\\&nbsp;&nbsp;\\simplify[std]{{p1}*{a[1][0]}+{q1}*{b[1][0]}+{r1}*{c[1][0]}}&amp;\\simplify[std]{{p1}*{a[1][1]}+{q1}*{b[1][1]}+{r1}*{c[1][1]}} \\end{pmatrix}\\\\ &amp;=&amp;\\simplify{{lcabc}}\\\\ &nbsp;\\end{eqnarray*} \\]</p>\n<p>&nbsp;</p>", "functions": {}, "extensions": ["stats"], "ungrouped_variables": ["a", "q1", "c", "b", "r1", "q", "p", "p1", "apb", "lcab", "lcabc"], "type": "question", "contributors": [{"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}]}]}], "contributors": [{"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}]}