// 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": [{"rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noleadingminus"]}, "functions": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "description": "", "definition": "matrix(repeat(repeat(random(-5..5 except 0),2),2))", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "description": "", "definition": "random(2..6)", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "description": "", "definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0]]),2),2))", "templateType": "anything"}, "p1": {"name": "p1", "group": "Ungrouped variables", "description": "", "definition": "random(2..6 except p)", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "description": "", "definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0],b[0][0]]),2),2))", "templateType": "anything"}, "lcabc": {"name": "lcabc", "group": "Ungrouped variables", "description": "", "definition": "p1*a+q1*b+r1*c", "templateType": "anything"}, "q": {"name": "q", "group": "Ungrouped variables", "description": "", "definition": "random(-6..6 except [0,1,-1,p])", "templateType": "anything"}, "lcab": {"name": "lcab", "group": "Ungrouped variables", "description": "", "definition": "p*a+q*b", "templateType": "anything"}, "apb": {"name": "apb", "group": "Ungrouped variables", "description": "", "definition": "a+b", "templateType": "anything"}, "q1": {"name": "q1", "group": "Ungrouped variables", "description": "", "definition": "random(-6..6 except [0,1,-1,p1,q])", "templateType": "anything"}, "r1": {"name": "r1", "group": "Ungrouped variables", "description": "", "definition": "random(-6..6 except [0,1,-1,p1,q1])", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "
Let
\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}},\\;\\; C=\\simplify{{c}}\\]
Calculate the following $2 \\times 2$ matrices:
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\\[ \\begin{eqnarray*} \\simplify[std]{A+B} &=&\\simplify[std]{{a}+{b}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{a[0][0]}+{b[0][0]}}& \\simplify[std]{{a[0][1]}+{b[0][1]}}\\\\ \\simplify[std]{{a[1][0]}+{b[1][0]}}&\\simplify[std]{{a[1][1]}+{b[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{apb}}\\\\ \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} \\simplify[std]{{p}A+{q}B} &=&\\simplify[std]{{p}{a}+{q}{b}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{p}*{a[0][0]}+{q}*{b[0][0]}}& \\simplify[std]{{p}*{a[0][1]}+{q}*{b[0][1]}}\\\\ \\simplify[std]{{p}*{a[1][0]}+{q}*{b[1][0]}}&\\simplify[std]{{p}*{a[1][1]}+{q}*{b[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{lcab}}\\\\ \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} \\simplify[std]{{p1}A+{q1}B+{r1}C} &=&\\simplify[std]{{p1}{a}+{q1}{b}+{r1}{c}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{p1}*{a[0][0]}+{q1}*{b[0][0]}+{r1}*{c[0][0]}}& \\simplify[std]{{p1}*{a[0][1]}+{q1}*{b[0][1]}+{r1}*{c[0][1]}}\\\\ \\simplify[std]{{p1}*{a[1][0]}+{q1}*{b[1][0]}+{r1}*{c[1][0]}}&\\simplify[std]{{p1}*{a[1][1]}+{q1}*{b[1][1]}+{r1}*{c[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{lcabc}}\\\\ \\end{eqnarray*} \\]
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Linear combinations of $2 \\times 2$ matrices. Three examples.
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