// Numbas version: exam_results_page_options
{"name": "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": [{"name": "Solve simultaneous equations by finding inverse matrix, ", "tags": ["checked2015", "inverse of a matrix", "Linear equations", "linear equations", "linear equations in matrix form", "matrices", "matrix", "matrix equations", "matrix form", "matrix multiplication", "multiply matrices", "multiply matrix", "solving linear equations", "system of linear equations", "tested1"], "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>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>", "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>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"mb": {"name": "mb", "group": "Ungrouped variables", "definition": "matrix([\n    [random(-9..9 except 0)],\n    [random(-9..9 except 0)]\n])", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(ma_inverse*mb)[0][0]", "description": "", "templateType": "anything", "can_override": false}, "ma": {"name": "ma", "group": "Ungrouped variables", "definition": "matrix([\n    [a00,a01],\n    [a10,a11]\n])", "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", "can_override": false}, "a10": {"name": "a10", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a00,-a00,a00*a11/a01])", "description": "", "templateType": "anything", "can_override": false}, "ma_inverse": {"name": "ma_inverse", "group": "Ungrouped variables", "definition": "matrix([\n    [ma[1][1], -ma[0][1]],\n    [-ma[1][0], ma[0][0]]\n])/det(ma)", "description": "", "templateType": "anything", "can_override": false}, "a01": {"name": "a01", "group": "Ungrouped variables", 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"nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the inverse of $\\mathbf{A}$.</p>\n<p>$\\mathbf{A}^{-1} = $&nbsp;[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "ma_inverse", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, 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