// Numbas version: finer_feedback_settings
{"name": "2x2 Matrix Problems", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "2x2 Matrix Problems", "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": "<h1>$2\\times2$ Matrix Problems</h1>\n<p>Solve the set of simultaneous equations using matrices:</p>", "advice": "<p><strong>a)</strong></p>\n<p>By using the <strong>coefficients</strong> of the variables in the equations, we can write the equations as:</p>\n<p>$$<br/>\\begin{pmatrix} \\var{a} &amp; \\var{b}\\\\ \\var{a1}&amp;\\var{b1} \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} \\var{c} \\\\ \\var{c1} \\end{pmatrix}<br/>$$</p>\n<p>Notice that carrying out the matrix multiplication will give you your system of equations.</p>\n<p><strong>b)</strong></p>\n<p>We first find the determinant of $\\boldsymbol{A}$:</p>\n<p>$$<br/>\\det \\boldsymbol{A} = \\var{a} \\times \\var{b1} - \\var{b} \\times \\var{a1} = \\var{det}<br/>$$</p>\n<p>Since $\\det \\boldsymbol{A} \\neq 0$, $A$ is invertible with:</p>\n<p>$$\\begin{aligned}<br/>A^{-1} &amp;= \\frac{1}{\\var{det}}\\begin{pmatrix}<br/>{\\var{b1}}&amp;{\\var{-b}} \\\\<br/>{\\var{-a1}}&amp;{\\var{a}}<br/>\\end{pmatrix} \\\\<br/>&amp;= \\begin{pmatrix}<br/>\\simplify[std]{{b1}/{det}}&amp;\\simplify[std]{{-b}/{det}} \\\\<br/>\\simplify[std]{{-a1}/{det}}&amp;\\simplify[std]{{a}/{det}} <br/>\\end{pmatrix}<br/>\\end{aligned}<br/>$$</p>\n<p><strong>c)</strong></p>\n<p>We have the equation:</p>\n<p>$$<br/>A^{-1}b = \\begin{pmatrix} <br/>\\simplify[std]{{b1}/{det}}&amp;\\simplify[std]{{-b}/{det}} \\\\<br/>\\simplify[std]{{-a1}/{det}}&amp;\\simplify[std]{{a}/{det}} <br/>\\end{pmatrix} \\begin{pmatrix} <br/>\\var{c} \\\\<br/>\\var{c1}<br/>\\end{pmatrix}<br/>$$</p>\n<p>Using standard matrix multiplication, we can compute:</p>\n<p>$$<br/>A^{-1}b = \\begin{pmatrix} <br/>\\simplify[std]{{b1}/{det}} \\times \\var{c} + \\simplify[std]{{-b}/{det}} \\times \\var{c1}\\\\<br/>\\simplify[std]{{-a1}/{det}} \\times \\var{c} + \\simplify[std]{{a}/{det}} \\times \\var{c1}<br/>\\end{pmatrix} = \\begin{pmatrix} \\simplify[std]{{c*b1-c1*b}/{det}}\\\\\\simplify[std]{{c1*a-c*a1}/{det}}\\end{pmatrix}&nbsp;<br/>$$</p>\n<p><strong>d)</strong></p>\n<p>Notice that</p>\n<p>$$<br/>\\boldsymbol{A}v = b \\Rightarrow v = \\boldsymbol{A}^{-1}b<br/>$$</p>\n<p>Hence, we can use our solution to part $c)$ to find:</p>\n<p>$$<br/>\\begin{pmatrix}&nbsp; <br/>x \\\\<br/>y <br/>\\end{pmatrix} = \\boldsymbol{A}^{-1}b =&nbsp; \\begin{pmatrix} <br/>\\simplify[std]{{c*b1-c1*b}/{det}} \\\\ <br/>\\simplify[std]{{c1*a-c*a1}/{det}}<br/>\\end{pmatrix}<br/>$$</p>\n<p>And therefore, we have:</p>\n<p>$$<br/>\\begin{eqnarray*} x&amp;=&amp; \\simplify[std]{{c*b1-c1*b}/{det}}\\\\ y&amp;=&amp; \\simplify[std]{{c1*a-c*a1}/{det}} \\end{eqnarray*} <br/>$$</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..5 )", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "det": {"name": "det", "group": "Ungrouped variables", "definition": "a*b1 - b*a1", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "det<>0", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "a1", "b1", "c1", "det"], "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: left;\">Write the following set of equations as a matrix equation:<br/><br/>$$<br/>\\begin{eqnarray*} \\simplify[std]{{a}x+{b}y}&amp;=&amp;\\var{c}\\\\ \\simplify[std]{{a1}x+{b1}y}&amp;=&amp;\\var{c1} \\end{eqnarray*} <br/>$$</p>\n<p style=\"text-align: left;\">That is, in the form:</p>\n<p style=\"text-align: left;\">$$<br/>\\boldsymbol{A}v=b<br/>$$</p>\n<p style=\"text-align: left;\">for some matrix $\\boldsymbol{A}$ and column vectors $v$ and $b$.</p>\n<p style=\"text-align: center;\"><br/>[[0]]$<br/>\\begin{pmatrix}<br/>x \\\\<br/>y \\\\<br/>\\end{pmatrix} = \\ $[[1]]&nbsp;</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([\n    [a,b],\n    [a1,b1]\n])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([\n    [c],\n    [c1]\n])", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"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><span>Find the inverse of $\\boldsymbol{A}$, the $2 \\times 2$ matrix you defined in the previous question</span></p>\n<p><em> Input all numbers as <strong>fractions or integers</strong> and not as decimals. Simplify your fractions as much as possible!</em></p>\n<p style=\"text-align: center;\">$\\boldsymbol{A}^{-1} = $ [[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([\n    [b1,-b],\n    [-a1,a]\n])/det", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"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>Using matrix multiplication and your answer to part $b)$, find $\\boldsymbol{A}^{-1}b$</p>\n<p><em> Give your answer as <strong>fractions or integers</strong> and not as decimals.</em></p>\n<p style=\"text-align: center;\">$\\boldsymbol{A}^{-1}b = $ [[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "1.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([{c*b1-c1*b}/{b1*a-a1*b}],[{-c*a1+c1*a}/{b1*a-a1*b}])", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"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: left;\">Using your answer to part $c)$, state the values of $x$ and $y$.</p>\n<p style=\"text-align: left;\"><em>Give your answer as <strong>fractions or integers</strong> and not as decimals.</em></p>\n<p style=\"text-align: center;\">$x = $ [[0]]</p>\n<p style=\"text-align: center;\">$y = $ [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 0.5, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{c*b1-c1*b}/{b1*a-a1*b}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input as a fraction or an integer, not as a decimal</p>"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 0.5, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{-c*a1+c1*a}/{b1*a-a1*b}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input as a fraction or an integer, not as a decimal</p>"}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}]}]}], "contributors": [{"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}]}