// Numbas version: exam_results_page_options
{"name": "Matrix Equations", "metadata": {"description": "<p>Try to solve some simultaneous equations using matrix inverses.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "questions": [{"name": "Alexander's copy of Solve simultaneous equations by finding inverse matrix,", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Alexander Corner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5328/"}], "variable_groups": [], "variables": {"mb": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([\n    [random(-9..9 except 0)],\n    [random(-9..9 except 0)]\n])", "description": "", "name": "mb"}, "x": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(ma_inverse*mb)[0][0]", "description": "", "name": "x"}, "ma": {"group": "Ungrouped variables", "templateType": "anything", "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>", "name": "ma"}, "a10": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except [0,a00,-a00,a00*a11/a01])", "description": "", "name": "a10"}, "ma_inverse": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([\n    [ma[1][1], -ma[0][1]],\n    [-ma[1][0], ma[0][0]]\n])/det(ma)", "description": "", "name": "ma_inverse"}, "a01": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "a01"}, "y": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(ma_inverse*mb)[1][0]", "description": "", "name": "y"}, "a11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except [0,a01,-a01])", "description": "", "name": "a11"}, "a00": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "a00"}}, "ungrouped_variables": ["ma", "a00", "a01", "a10", "a11", "mb", "ma_inverse", "x", "y"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"allowFractions": true, "correctAnswer": "ma", "showCorrectAnswer": true, "allowResize": false, "correctAnswerFractions": false, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "markPerCell": false, "marks": 1}, {"answer": "x", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": "0.25", "vsetrangepoints": 5}, {"answer": "y", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": "0.25", "vsetrangepoints": 5}, {"allowFractions": true, "correctAnswer": "mb", "showCorrectAnswer": true, "allowResize": false, "correctAnswerFractions": false, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "marks": "0.5"}], "type": "gapfill", "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>", "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": true, "correctAnswer": "ma_inverse", "showCorrectAnswer": true, "allowResize": false, "correctAnswerFractions": true, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "markPerCell": false, "marks": "2"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Find the inverse of $\\mathbf{A}$.</p>\n<p>$\\mathbf{A}^{-1} = $&nbsp;[[0]]</p>", "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": true, "correctAnswer": "ma_inverse*mb", "showCorrectAnswer": true, "allowResize": false, "correctAnswerFractions": true, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Now find $\\mathbf{A}^{-1}\\mathbf{b}$.</p>\n<p><span>$\\mathbf{A}^{-1}\\mathbf{b} = $&nbsp;[[0]]</span></p>", "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": true, "scripts": {}, "type": "numberentry", "correctAnswerFraction": true, "minValue": "x", "maxValue": "x", "marks": "0.5"}, {"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": true, "scripts": {}, "type": "numberentry", "correctAnswerFraction": true, "minValue": "y", "maxValue": "y", "marks": "0.5"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Finally, solve the equations.</p>\n<p>$x = $&nbsp;[[0]]</p>\n<p>$y = $&nbsp;[[1]]</p>", "marks": 0}], "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>", "tags": ["checked2015", "inverse of a matrix", "linear equations", "linear equations in matrix form", "MAS1602", "MAS2223", "matrices", "matrix", "matrix equations", "matrix form", "matrix multiplication", "multiply matrices", "multiply matrix", "solving linear equations", "system of linear equations", "tested1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>20/06/2012:</strong></p>\n    <p>Added, edited tags.</p>\n    <p>Edited advice so that it gave the correct solution for $y$ (as in the answer).</p>\n    <p><strong>4/07/2012:<br /></strong></p>\n    <p>Column vectors v and b have the bracket in the incorrect place.</p>\n    <p>&nbsp;</p>\n    <p><strong>10/07/2012:<br /></strong></p>\n    <p>Added tags.<strong><br /></strong></p>\n    <p>Question appears to be working correctly.</p>\n    <p>Column vectors v and b still have brackets in incorrect places.</p>\n    <p><strong>24/12/2012:</strong></p>\n    <p>Checked calculations, OK. Added tested1 tag.</p>\n    <p>Improved display as requested above.</p>", "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>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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>"}]}], "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "contributors": [{"name": "Alexander Corner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5328/"}], "extensions": [], "custom_part_types": [], "resources": []}