// Numbas version: exam_results_page_options
{"name": "Inverse Matrix Method (2x2)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Inverse Matrix Method (2x2)", "tags": [], "metadata": {"description": "<p>Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Write the following equations as a matrix equation <br/>\\[Av=b\\]for a matrix $A$ and column vectors $v$ and $b$<br/>\\[ \\begin{eqnarray*} \\simplify[std]{{a}x+{b}y}&amp;=&amp;\\var{c}\\\\ \\simplify[std]{{a1}x+{b1}y}&amp;=&amp;\\var{c1} \\end{eqnarray*} \\]</p>", "advice": "<h4>a)</h4>\n<p>The equations can be written in the matrix form:</p>\n<p>\\[\\begin{pmatrix} \\var{a} &amp; \\var{b}\\\\ \\var{a1}&amp;\\var{b1} \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} \\var{c} \\\\ \\var{c1} \\end{pmatrix}\\]</p>\n<h4>b)</h4>\n<p>Since $\\mathrm{det}(A) = \\simplify[]{{a}*{b1}-{b}*{a1}={dA}} \\neq 0$, $A$ is invertible and</p>\n<p>\\[A^{-1} = \\begin{pmatrix} \\simplify[std]{{b1}/{dA}}&amp;\\simplify[std]{{-b}/{dA}}\\\\\\simplify[std]{{-a1}/{dA}}&amp;\\simplify[std]{{a}/{dA}} \\end{pmatrix}\\]</p>\n<h4>c)</h4>\n<p>We have:</p>\n<p>\\[ \\begin{eqnarray*} A^{-1}b &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{b1}/{dA}}&amp;\\simplify[std]{{-b}/{dA}}\\\\\\simplify[std]{{-a1}/{dA}}&amp;\\simplify[std]{{a}/{dA}} \\end{pmatrix}\\begin{pmatrix} \\var{c}\\\\\\var{c1}\\end{pmatrix} \\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{c*b1-c1*b}/{dA}}\\\\\\simplify[std]{{c1*a-c*a1}/{dA}}\\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>d)</h4>\n<p>Note that $Av = b \\Rightarrow v = A^{-1}b$ hence we can read the solution from the last part as this gives:</p>\n<p>\\[\\begin{pmatrix} x\\\\y \\end{pmatrix} = \\begin{pmatrix} \\simplify[std]{{c*b1-c1*b}/{dA}}\\\\ \\simplify[std]{{c1*a-c*a1}/{dA}}\\end{pmatrix}\\]</p>\n<p>Hence \\[\\begin{eqnarray*} x&amp;=&amp; \\simplify[std]{{c*b1-c1*b}/{dA}}\\\\ y&amp;=&amp; \\simplify[std]{{c1*a-c*a1}/{dA}} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"sa": {"name": "sa", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "sa1": {"name": "sa1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "sb*random(1..9)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "sc*random(1..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "sa*random(1..9)", "description": "", "templateType": "anything"}, "sb": {"name": "sb", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "addortakeaway": {"name": "addortakeaway", "group": "Ungrouped variables", "definition": "if(b*b1<0,'add the equation','take away the equation')", "description": "", "templateType": "anything"}, "sc": {"name": "sc", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "test": {"name": "test", "group": "Ungrouped variables", "definition": "round(a*b1/b)", "description": "", "templateType": "anything"}, "ab": {"name": "ab", "group": "Ungrouped variables", "definition": "abs(b)", "description": "", "templateType": "anything"}, "inc": {"name": "inc", "group": "Ungrouped variables", "definition": "sa*random(1..9)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "if(test+inc=0,test+inc+1,test+inc)", "description": "", "templateType": "anything"}, "sc1": {"name": "sc1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "da": {"name": "da", "group": "Ungrouped variables", "definition": "a*b1-a1*b", "description": "", "templateType": "anything"}, "sb1": {"name": "sb1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "if(ab=1,sb1*random(2..9),if(ab=2 or ab=4 or ab=8,sb1*random(1,3,5,7,9), if(ab=3 or ab=6 or ab=9,sb1*random(1,2,4,5,6,8),if(ab=5,sb1*random(1,2,3,4,6,7,8,9),sb1*random(1,2,3,4,5,6,8,9)))))", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "sc1*random(1..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "addortakeaway", "c", "b", "da", "a1", "sc", "ab", "sb1", "b1", "test", "sb", "sa", "sc1", "sa1", "c1", "inc"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: center;\">$A = $&nbsp;[[0]]</p>\n<table style=\"margin-left: auto; 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