// Numbas version: exam_results_page_options
{"name": "W1b - Solve simultaneous equations by finding inverse matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "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"}, "name": "W1b - Solve simultaneous equations by finding inverse matrix", "functions": {}, "statement": "<p>Rewrite the following system of equations as a matrix equation</p>\n<p>\\[ \\mathbf{Av} = \\mathbf{b} \\]</p>\n<p>for a matrix $\\mathbf{A}$ and column vectors $\\mathbf{v}$ and $\\mathbf{b}$.</p>\n<p>\\begin{align}<br/>\\simplify[std]{ {ma[0][0]}x + {ma[0][1]}y} &amp;= \\var{mb[0][0]} \\\\<br/>\\simplify[std]{ {ma[1][0]}x + {ma[1][1]}y} &amp;= \\var{mb[1][0]}<br/>\\end{align}</p>\n<p><em><span>Input all numbers as fractions or integers and not as decimals.</span></em></p>", "parts": [{"variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "gaps": [{"correctAnswerFractions": false, "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "correctAnswer": "ma", "numRows": "2", "type": "matrix", "unitTests": [], "showCorrectAnswer": true, "variableReplacements": [], "adaptiveMarkingPenalty": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "markPerCell": true, "marks": "4", "customName": "", "tolerance": 0, "allowResize": false, "numColumns": "2"}, {"extendBaseMarkingAlgorithm": true, "type": "jme", "unitTests": [], "showCorrectAnswer": true, "checkVariableNames": false, "variableReplacements": [], "useCustomName": false, "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "answer": "x", "valuegenerators": [{"value": "", "name": "x"}], "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "showPreview": false, "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "marks": "1", "customName": "", "vsetRange": [0, 1], "checkingAccuracy": 0.001, "scripts": {}}, {"extendBaseMarkingAlgorithm": true, "type": "jme", "unitTests": [], "showCorrectAnswer": true, "checkVariableNames": false, "variableReplacements": [], "useCustomName": false, "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "answer": "y", "valuegenerators": [{"value": "", "name": "y"}], "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "showPreview": false, "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "marks": "1", "customName": "", "vsetRange": [0, 1], "checkingAccuracy": 0.001, "scripts": {}}, {"correctAnswerFractions": false, "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "correctAnswer": "mb", "numRows": "2", "type": "matrix", "unitTests": [], "showCorrectAnswer": true, "variableReplacements": [], "adaptiveMarkingPenalty": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "markPerCell": true, "marks": "2", "customName": "", "tolerance": 0, "allowResize": false, "numColumns": 1}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "marks": 0, "customName": "", "unitTests": [], "prompt": "<p style=\"text-align: center;\">$\\mathbf{A} = $&nbsp;[[0]]</p>\n<div style=\"text-align: center;\">$\\mathbf{v} = $&nbsp;\n<div class=\"matrix-input\">\n<div class=\"matrix-wrapper\"><span class=\"left-bracket\"></span>\n<table class=\"matrix\">\n<tbody>\n<tr>\n<td>[[1]]</td>\n</tr>\n<tr>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>\n<span class=\"right-bracket\"></span></div>\n</div>\n<p style=\"text-align: center;\">$\\mathbf{b} = $ [[3]]</p>\n</div>"}, {"variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "gaps": [{"correctAnswerFractions": true, "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "correctAnswer": "ma_inverse", "numRows": "2", "type": "matrix", "unitTests": [], "showCorrectAnswer": true, "variableReplacements": [], "adaptiveMarkingPenalty": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "markPerCell": true, "marks": "4", "customName": "", "tolerance": 0, "allowResize": false, "numColumns": "2"}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "marks": 0, "customName": "", "unitTests": [], "prompt": "<p>Find the inverse of $\\mathbf{A}$.</p>\n<p>$\\mathbf{A}^{-1} = $&nbsp;[[0]]</p>"}, {"variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "gaps": [{"correctAnswerFractions": true, "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "correctAnswer": "ma_inverse*mb", "numRows": "2", "type": "matrix", "unitTests": [], "showCorrectAnswer": true, "variableReplacements": [], "adaptiveMarkingPenalty": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "markPerCell": true, "marks": "2", "customName": "", "tolerance": 0, "allowResize": false, "numColumns": 1}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "marks": 0, "customName": "", "unitTests": [], "prompt": "<p>Now find $\\mathbf{v} = \\mathbf{A}^{-1}\\mathbf{b}$.</p>\n<p><span>$\\mathbf{A}^{-1}\\mathbf{b} = $&nbsp;[[0]]</span></p>"}], "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "<h4>a)</h4>\n<p>The equations can be written in the matrix form</p>\n<p>\\[ \\var{ma}\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\var{mb} \\]</p>\n<h4><strong></strong>b)</h4>\n<p>$\\mathrm{det}(\\mathbf{A}) = \\simplify[]{ {ma[0][0]}*{ma[1][1]} - {ma[0][1]}*{ma[1][0]}} = \\var{det(ma)} \\neq 0$, so $\\mathbf{A}$ is invertible.</p>\n<p>\\[ \\mathbf{A}^{-1} = \\simplify[fractionnumbers]{{ma_inverse}} \\]</p>\n<h4>c)</h4>\n<p>We have&nbsp;</p>\n<p>\\begin{align}<br/>\\mathbf{A}^{-1}\\mathbf{b} &amp;= \\simplify[fractionnumbers]{{ma_inverse}*{mb}} \\\\<br/>&amp;= \\simplify[fractionnumbers]{{ma_inverse*mb}}<br/>\\end{align}</p>\n<h4>d)</h4>\n<p>Rearrange the equation $\\mathbf{Av}=\\mathbf{b}$ to make $\\mathbf{v}$ the subject:</p>\n<p>\\begin{align}<br/>\\mathbf{A}^{-1}\\mathbf{A}\\mathbf{v} &amp;= \\mathbf{A}^{-1}\\mathbf{b} \\\\<br/>\\mathbf{v} &amp;= \\mathbf{A}^{-1}\\mathbf{b} \\\\ \\\\<br/>\\end{align}</p>\n<p>Hence,</p>\n<p>\\[ \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\simplify[fractionnumbers]{{ma_inverse*mb}} \\]</p>\n<p>That is,</p>\n<p>\\begin{align}<br/>x &amp;= \\simplify[fractionnumbers]{{x}}, \\\\ \\\\<br/>y &amp;= \\simplify[fractionnumbers]{{y}}<br/>\\end{align}</p>", "extensions": [], "variables": {"a11": {"description": "", "templateType": "anything", "definition": "random(-9..9 except [0,a01,-a01])", "group": "Ungrouped variables", "name": "a11"}, "mb": {"description": "", "templateType": "anything", "definition": "matrix([\n    [random(-9..9 except 0)],\n    [random(-9..9 except 0)]\n])", "group": "Ungrouped variables", "name": "mb"}, "ma": {"description": "<p>Matrix A. a10 is picked so it's non-singular, and a11 is never $\\pm a01$.</p>\n<p>No entry is 0.</p>", "templateType": "anything", "definition": "matrix([\n    [a00,a01],\n    [a10,a11]\n])", "group": "Ungrouped variables", "name": "ma"}, "a10": {"description": "", "templateType": "anything", "definition": "random(-9..9 except [0,a00,-a00,a00*a11/a01])", "group": "Ungrouped variables", "name": "a10"}, "x": {"description": "", "templateType": "anything", "definition": "(ma_inverse*mb)[0][0]", "group": "Ungrouped variables", "name": "x"}, "y": {"description": "", "templateType": "anything", "definition": "(ma_inverse*mb)[1][0]", "group": "Ungrouped variables", "name": "y"}, "a00": {"description": "", "templateType": "anything", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "name": "a00"}, "a01": {"description": "", "templateType": "anything", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "name": "a01"}, "ma_inverse": {"description": "", "templateType": "anything", "definition": "matrix([\n    [ma[1][1], -ma[0][1]],\n    [-ma[1][0], ma[0][0]]\n])/det(ma)", "group": "Ungrouped variables", "name": "ma_inverse"}}, "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["ma", "a00", "a01", "a10", "a11", "mb", "ma_inverse", "x", "y"], "variable_groups": [], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}