// Numbas version: exam_results_page_options
{"name": "Simon's copy of 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": [{"extensions": [], "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix.&nbsp;</p>"}, "ungrouped_variables": ["ma", "a00", "a01", "a10", "a11", "mb", "ma_inverse", "x", "y"], "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>", "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "parts": [{"showCorrectAnswer": true, "prompt": "<p style=\"text-align: center;\">$\\mathbf{A} = $&nbsp;[[0]]</p>\n<div style=\"text-align: center;\">$\\mathbf{v} = $&nbsp;\n<div class=\"matrix-input\">\n  <div class=\"matrix-wrapper\">\n    <span class=\"left-bracket\"></span>\n    <table class=\"matrix\">\n      <tbody>\n        <tr>\n          <td>[[1]]</td>\n        </tr>\n        <tr>\n          <td>[[2]]</td>\n        </tr>\n      </tbody>\n    </table>\n    <span class=\"right-bracket\"></span>\n  </div>\n</div>\n<p style=\"text-align: center;\">$\\mathbf{b} = $ [[3]]</p></div>", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false, "type": "gapfill", "scripts": {}, "marks": 0, "unitTests": [], "gaps": [{"showCorrectAnswer": true, "numRows": "2", "allowResize": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "type": "matrix", "tolerance": 0, "correctAnswer": "ma", "scripts": {}, "marks": 1, "markPerCell": false, "unitTests": [], "correctAnswerFractions": false, "allowFractions": true, "numColumns": "2", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}, {"showCorrectAnswer": true, "expectedVariableNames": [], "showPreview": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "type": "jme", "checkingAccuracy": 0.001, "answer": "x", "checkVariableNames": false, "failureRate": 1, "scripts": {}, "marks": "0.25", "vsetRange": [0, 1], "unitTests": [], "vsetRangePoints": 5, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}, {"showCorrectAnswer": true, "expectedVariableNames": [], "showPreview": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "type": "jme", "checkingAccuracy": 0.001, "answer": "y", "checkVariableNames": false, "failureRate": 1, "scripts": {}, "marks": "0.25", "vsetRange": [0, 1], "unitTests": [], "vsetRangePoints": 5, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}, {"showCorrectAnswer": true, "numRows": "2", "allowResize": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "type": "matrix", "tolerance": 0, "correctAnswer": "mb", "scripts": {}, "marks": "0.5", "markPerCell": false, "unitTests": [], "correctAnswerFractions": false, "allowFractions": true, "numColumns": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}], "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}, {"showCorrectAnswer": true, "prompt": "<p>Find the inverse of $\\mathbf{A}$.</p>\n<p>$\\mathbf{A}^{-1} = $&nbsp;[[0]]</p>", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false, "type": "gapfill", "scripts": {}, "marks": 0, "unitTests": [], "gaps": [{"showCorrectAnswer": true, "numRows": "2", "allowResize": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "type": "matrix", "tolerance": 0, "correctAnswer": "ma_inverse", "scripts": {}, "marks": "2", "markPerCell": false, "unitTests": [], "correctAnswerFractions": true, "allowFractions": true, "numColumns": "2", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}], "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}, {"showCorrectAnswer": true, "prompt": "<p>Now find $\\mathbf{A}^{-1}\\mathbf{b}$.</p>\n<p><span>$\\mathbf{A}^{-1}\\mathbf{b} = $&nbsp;[[0]]</span></p>", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false, "type": "gapfill", "scripts": {}, "marks": 0, "unitTests": [], "gaps": [{"showCorrectAnswer": true, "numRows": "2", "allowResize": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "type": "matrix", "tolerance": 0, "correctAnswer": "ma_inverse*mb", "scripts": {}, "marks": 1, "markPerCell": false, "unitTests": [], "correctAnswerFractions": true, "allowFractions": true, "numColumns": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}], "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}, {"showCorrectAnswer": true, "prompt": "<p>Finally, solve the equations.</p>\n<p>$x = $&nbsp;[[0]]</p>\n<p>$y = $&nbsp;[[1]]</p>", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false, "type": "gapfill", "scripts": {}, "marks": 0, "unitTests": [], "gaps": [{"showCorrectAnswer": true, "minValue": "x", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "type": "numberentry", "mustBeReduced": false, "correctAnswerFraction": true, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain", "unitTests": [], "maxValue": "x", "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}, {"showCorrectAnswer": true, "minValue": "y", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "type": "numberentry", "mustBeReduced": false, "correctAnswerFraction": true, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain", "unitTests": [], "maxValue": "y", "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}], "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}], "name": "Simon's copy of Solve simultaneous equations by finding inverse matrix,", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "preamble": {"js": "", "css": ""}, "variables": {"mb": {"definition": "matrix([\n    [random(-9..9 except 0)],\n    [random(-9..9 except 0)]\n])", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "mb"}, "a01": {"definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a01"}, "x": {"definition": "(ma_inverse*mb)[0][0]", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "x"}, "a10": {"definition": "random(-9..9 except [0,a00,-a00,a00*a11/a01])", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a10"}, "ma_inverse": {"definition": "matrix([\n    [ma[1][1], -ma[0][1]],\n    [-ma[1][0], ma[0][0]]\n])/det(ma)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ma_inverse"}, "a00": {"definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a00"}, "ma": {"definition": "matrix([\n    [a00,a01],\n    [a10,a11]\n])", "group": "Ungrouped variables", "templateType": "anything", "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>", "name": "ma"}, "y": {"definition": "(ma_inverse*mb)[1][0]", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "y"}, "a11": {"definition": "random(-9..9 except [0,a01,-a01])", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a11"}}, "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} = \\frac{1}{\\var{det(ma)}} \\begin{pmatrix} \\var{ma[1][1]}&amp;\\var{-ma[0][1]} \\\\ \\var{-ma[1][0]}&amp;\\var{ma[0][0]} \\end{pmatrix}&nbsp; = \\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>Given the equation $\\mathbf{Av}=\\mathbf{b}$ we can make $\\mathbf{v}$ the subject by pre-multiplying each side by $\\mathbf{A}^{-1}$:</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>", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}