// Numbas version: finer_feedback_settings
{"name": "Mark's copy of Mark's copy of Inverse of a 3x3 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b21": {"definition": "(a31*a23-a21*a33)/a11", "templateType": "anything", "name": "b21", "group": "Ungrouped variables", "description": ""}, "b33": {"definition": "(a11*a22-a12*a21)/a11", "templateType": "anything", "name": "b33", "group": "Ungrouped variables", "description": ""}, "a31": {"definition": "k1*a11", "templateType": "anything", "name": "a31", "group": "Ungrouped variables", "description": ""}, "b11": {"definition": "(a22*a33-a32*a23)/a11", "templateType": "anything", "name": "b11", "group": "Ungrouped variables", "description": ""}, "a12": {"definition": "random(2..6#1)", "templateType": "randrange", "name": "a12", "group": "Ungrouped variables", "description": ""}, "t1": {"definition": "random(5..12#1)", "templateType": "randrange", "name": "t1", "group": "Ungrouped variables", "description": ""}, "a33": {"definition": "{k1}*({a23}+{a13}-{k}*{a13})+1", "templateType": "anything", "name": "a33", "group": "Ungrouped variables", "description": "<p>3x33 matrix with determinant = a11</p>"}, "a13": {"definition": "random(0..10#1)", "templateType": "randrange", "name": "a13", "group": "Ungrouped variables", "description": ""}, "a32": {"definition": "k1*(a12+1)", "templateType": "anything", "name": "a32", "group": "Ungrouped variables", "description": ""}, "r": {"definition": "a11*r1+a12*s1+a13*t1", "templateType": "anything", "name": "r", "group": "Ungrouped variables", "description": ""}, "a11": {"definition": "random(1,2,4,5,10)", "templateType": "anything", "name": "a11", "group": "Ungrouped variables", "description": ""}, "t": {"definition": "a31*r1+a32*s1+a33*t1", "templateType": "anything", "name": "t", "group": "Ungrouped variables", "description": ""}, "a22": {"definition": "k*a12+1", "templateType": "anything", "name": "a22", "group": "Ungrouped variables", "description": ""}, "s1": {"definition": "random(0..5#1)", "templateType": "randrange", "name": "s1", "group": "Ungrouped variables", "description": ""}, "k": {"definition": "random(2..5#1)", "templateType": "randrange", "name": "k", "group": "Ungrouped variables", "description": ""}, "a23": {"definition": "random(1..8#1)", "templateType": "randrange", "name": "a23", "group": "Ungrouped variables", "description": ""}, "b13": {"definition": "(a12*a23-a22*a13)/a11", "templateType": "anything", "name": "b13", "group": "Ungrouped variables", "description": ""}, "k1": {"definition": "random(2..6#1)", "templateType": "randrange", "name": "k1", "group": "Ungrouped variables", "description": ""}, "s": {"definition": "a21*r1+a22*s1+a23*t1", "templateType": "anything", "name": "s", "group": "Ungrouped variables", "description": ""}, "b12": {"definition": "(a32*a13-a12*a33)/a11", "templateType": "anything", "name": "b12", "group": "Ungrouped variables", "description": ""}, "r1": {"definition": "random(1..6#1)", "templateType": "randrange", "name": "r1", "group": "Ungrouped variables", "description": ""}, "a21": {"definition": "k*a11", "templateType": "anything", "name": "a21", "group": "Ungrouped variables", "description": ""}, "b22": {"definition": "(a11*a33-a31*a13)/a11", "templateType": "anything", "name": "b22", "group": "Ungrouped variables", "description": ""}, "b31": {"definition": "(a21*a32-a22*a31)/a11", "templateType": "anything", "name": "b31", "group": "Ungrouped variables", "description": ""}, "b23": {"definition": "(a21*a13-a11*a23)/a11", "templateType": "anything", "name": "b23", "group": "Ungrouped variables", "description": ""}, "b32": {"definition": "(a12*a31-a11*a32)/a11", "templateType": "anything", "name": "b32", "group": "Ungrouped variables", "description": ""}}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "parts": [{"gaps": [{"scripts": {}, "showCorrectAnswer": true, "maxValue": "{a11}", "variableReplacements": [], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "marks": 1, "strictPrecision": false, "showPrecisionHint": false, "precisionType": "dp", "precision": 0, "type": "numberentry", "allowFractions": false, "minValue": "{a11}", "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision."}, {"precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "allowResize": true, "correctAnswerFractions": false, "showFeedbackIcon": true, "markPerCell": false, "marks": "4.5", "strictPrecision": false, "precisionType": "dp", "precision": "2", "correctAnswer": "matrix([\n  [b11,b12,b13],\n  [b21,b22,b23],\n  [b31,b32,b33]\n ]) ", "type": "matrix", "allowFractions": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "numColumns": "3", "variableReplacements": [], "tolerance": 0, "numRows": "3", "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showCorrectAnswer": true, "maxValue": "r1", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "minValue": "r1", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": []}, {"scripts": {}, "showCorrectAnswer": true, "maxValue": "s1", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "minValue": "s1", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": []}, {"scripts": {}, "showCorrectAnswer": true, "maxValue": "t1", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "minValue": "t1", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": []}], "prompt": "<p>Calculate the determinant of the matrix.</p>\n<p style=\"padding-left: 30px;\">\\(|A|=\\)&nbsp;[[0]]</p>\n<p>Calculate the inverse matrix and input the entries correct to two decimal places.</p>\n<p style=\"padding-left: 30px;\">\\(A^{-1}=\\)&nbsp;[[1]]</p>\n<p>Hence solve the following system of equations.</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{23}\\\\ \\var{a31}&amp;\\var{a32}&amp;\\var{a33}\\end{pmatrix}\\begin{pmatrix} x\\\\y\\\\z\\end{pmatrix}=\\begin{pmatrix} \\var{r}\\\\\\var{s}\\\\\\var{t}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 90px;\">\\(x=\\)&nbsp;[[2]]</p>\n<p style=\"padding-left: 90px;\">\\(y=\\)&nbsp;[[3]]</p>\n<p style=\"padding-left: 90px;\">\\(z=\\)&nbsp;[[4]]</p>\n<p></p>", "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0}], "statement": "<p>Given the matrix:</p>\n<p style=\"padding-left: 30px;\">\\(\\mathbf{A}=\\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}\\\\\\var{a31}&amp;\\var{a32}&amp;\\var{a33} \\end{pmatrix}\\)</p>", "tags": [], "advice": "<p>Given the matrix:</p>\n<p style=\"padding-left: 30px;\">\\(\\mathbf{A}=\\begin{pmatrix} {a11}&amp;{a12}&amp;{a13}\\\\ {a21}&amp;{a22}&amp;{a23}\\\\{a31}&amp;{a32}&amp;{a33} \\end{pmatrix}\\)</p>\n<p>The determinant of a 3x3 matrix is determined by the formula:</p>\n<p style=\"padding-left: 30px;\">\\(|\\mathbf{A}|={a11}*\\begin{vmatrix}{a22}&amp;{a23}\\\\{a32}&amp;{a33}\\end{vmatrix}-{a12}*\\begin{vmatrix}{a21}&amp;{a23}\\\\{a31}&amp;{a33}\\end{vmatrix}+{a13}*\\begin{vmatrix}{a21}&amp;{a22}\\\\{a31}&amp;{a32}\\end{vmatrix}\\)</p>\n<p>So in this example:</p>\n<p style=\"padding-left: 30px;\">\\(\\mathbf{A}=\\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}\\\\\\var{a31}&amp;\\var{a32}&amp;\\var{a33} \\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(|\\mathbf{A}|=\\var{a11}*\\begin{vmatrix}\\var{a22}&amp;\\var{a23}\\\\\\var{a32}&amp;\\var{a33}\\end{vmatrix}-\\var{a12}*\\begin{vmatrix}\\var{a21}&amp;\\var{a23}\\\\\\var{a31}&amp;\\var{a33}\\end{vmatrix}+\\var{a13}*\\begin{vmatrix}\\var{a21}&amp;\\var{a22}\\\\\\var{a31}&amp;\\var{a32}\\end{vmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(|\\mathbf{A}|=\\var{a11}*(\\var{a22}*\\var{a33}-\\var{a32}*\\var{a23})-\\var{a12}*(\\var{a21}*\\var{a33}-\\var{a31}*\\var{a23})+\\var{a13}*(\\var{a21}*\\var{a32}-\\var{a31}*\\var{a22})\\)</p>\n<p style=\"padding-left: 30px;\">\\(|\\mathbf{A}|=(\\simplify{{a11}*({a22}*{a33}-{a32}*{a23})})-(\\simplify{{a12}*({a21}*{a33}-{a31}*{a23})})+(\\simplify{{a13}*({a21}*{a32}-{a31}*{a22})})\\)</p>\n<p style=\"padding-left: 30px;\">\\(|\\mathbf{A}|=\\var{a11}\\)</p>\n<p>The adjoint of the matrix is given by:</p>\n<p style=\"padding-left: 30px;\">\\(\\mathbf{A}^*=\\begin{pmatrix}\\begin{vmatrix}\\var{a22}&amp;\\var{a23}\\\\\\var{a32}&amp;\\var{a33}\\end{vmatrix}&amp;-\\begin{vmatrix}\\var{a21}&amp;\\var{a23}\\\\\\var{a31}&amp;\\var{a33}\\end{vmatrix}&amp;\\begin{vmatrix}\\var{a21}&amp;\\var{a22}\\\\\\var{a31}&amp;\\var{a32}\\end{vmatrix}\\\\-\\begin{vmatrix}\\var{a12}&amp;\\var{a13}\\\\\\var{a32}&amp;\\var{a33}\\end{vmatrix}&amp;\\begin{vmatrix}\\var{a11}&amp;\\var{a13}\\\\\\var{a31}&amp;\\var{a33}\\end{vmatrix}&amp;-\\begin{vmatrix}\\var{a11}&amp;\\var{a12}\\\\\\var{a31}&amp;\\var{a32}\\end{vmatrix}\\\\\\begin{vmatrix}\\var{a12}&amp;\\var{a13}\\\\\\var{a22}&amp;\\var{a23}\\end{vmatrix}&amp;-\\begin{vmatrix}\\var{a11}&amp;\\var{a13}\\\\\\var{a21}&amp;\\var{a23}\\end{vmatrix}&amp;\\begin{vmatrix}\\var{a11}&amp;\\var{a12}\\\\\\var{a21}&amp;\\var{a11}\\end{vmatrix}\\end{pmatrix}^t\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\mathbf{A}^*=\\begin{pmatrix}\\simplify{({a22}*{a33}-{a32}*{a23})}&amp;\\simplify{({a23}*{a31}-{a21}*{a33})}&amp;\\simplify{({a21}*{a32}-{a22}*{a31})}\\\\\\simplify{(-{a12}*{a33}+{a32}*{a13})}&amp;\\simplify{({a11}*{a33}-{a13}*{a31})}&amp;\\simplify{({a12}*{a31}-{a32}*{a11})}\\\\\\simplify{({a12}*{a23}-{a22}*{a13})}&amp;\\simplify{({a21}*{a13}-{a11}*{a23})}&amp;\\simplify{({a22}*{a11}-{a12}*{a21})}\\end{pmatrix}^t\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\mathbf{A}^*=\\begin{pmatrix}\\simplify{({a22}*{a33}-{a32}*{a23})}&amp;\\simplify{(-{a12}*{a33}+{a32}*{a13})}&amp;\\simplify{({a12}*{a23}-{a22}*{a13})}\\\\\\simplify{({a23}*{a31}-{a21}*{a33})}&amp;\\simplify{({a11}*{a33}-{a13}*{a31})}&amp;\\simplify{({a21}*{a13}-{a11}*{a23})}\\\\\\simplify{({a21}*{a32}-{a22}*{a31})}&amp;\\simplify{({a12}*{a31}-{a32}*{a11})}&amp;\\simplify{({a22}*{a11}-{a12}*{a21})}\\end{pmatrix}\\)</p>\n<p>The inverse matrix is defined by \\(A^{-1}=\\frac{1}{|A|}A^*\\)</p>\n<p style=\"padding-left: 30px;\">\\(A^{-1}=\\frac{1}{\\var{a11}}\\begin{pmatrix}\\simplify{({a22}*{a33}-{a32}*{a23})}&amp;\\simplify{(-{a12}*{a33}+{a32}*{a13})}&amp;\\simplify{({a12}*{a23}-{a22}*{a13})}\\\\\\simplify{({a23}*{a31}-{a21}*{a33})}&amp;\\simplify{({a11}*{a33}-{a13}*{a31})}&amp;\\simplify{({a21}*{a13}-{a11}*{a23})}\\\\\\simplify{({a21}*{a32}-{a22}*{a31})}&amp;\\simplify{({a12}*{a31}-{a32}*{a11})}&amp;\\simplify{({a22}*{a11}-{a12}*{a21})}\\end{pmatrix}\\)</p>\n<p>Using the inverse matrix method gives:</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{pmatrix} x\\\\y\\\\z\\end{pmatrix}=\\frac{1}{\\var{a11}}\\begin{pmatrix}\\simplify{({a22}*{a33}-{a32}*{a23})}&amp;\\simplify{(-{a12}*{a33}+{a32}*{a13})}&amp;\\simplify{({a12}*{a23}-{a22}*{a13})}\\\\\\simplify{({a23}*{a31}-{a21}*{a33})}&amp;\\simplify{({a11}*{a33}-{a13}*{a31})}&amp;\\simplify{({a21}*{a13}-{a11}*{a23})}\\\\\\simplify{({a21}*{a32}-{a22}*{a31})}&amp;\\simplify{({a12}*{a31}-{a32}*{a11})}&amp;\\simplify{({a22}*{a11}-{a12}*{a21})}\\end{pmatrix}\\begin{pmatrix} \\var{r}\\\\\\var{s}\\\\\\var{t}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{pmatrix} x\\\\y\\\\z\\end{pmatrix}=\\begin{pmatrix}\\var{b11}&amp;\\var{b12}&amp;\\var{b13}\\\\\\var{b21}&amp;\\var{b22}&amp;\\var{b23}\\\\\\var{b31}&amp;\\var{b32}&amp;\\var{b33}\\end{pmatrix}\\begin{pmatrix} \\var{r}\\\\\\var{s}\\\\\\var{t}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{pmatrix} x\\\\y\\\\z\\end{pmatrix}=\\begin{pmatrix}(\\var{b11})*\\var{r}+(\\var{b12})*\\var{s}+(\\var{b13})*\\var{t}\\\\(\\var{b21})*\\var{r}+(\\var{b22})*\\var{s}+(\\var{b23})*\\var{t}\\\\(\\var{b31})*\\var{r}+(\\var{b32})*\\var{s}+(\\var{b33})*\\var{t}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{pmatrix} x\\\\y\\\\z\\end{pmatrix}=\\begin{pmatrix} \\var{r1}\\\\\\var{s1}\\\\\\var{t1}\\end{pmatrix}\\)</p>", "ungrouped_variables": ["a11", "a12", "a13", "k", "a21", "a22", "a23", "k1", "a31", "a32", "a33", "b12", "b13", "b21", "b22", "b23", "b31", "b32", "b33", "b11", "r1", "s1", "t1", "r", "s", "t"], "rulesets": {}, "variable_groups": [], "name": "Mark's copy of Mark's copy of Inverse of a 3x3 matrix", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>This question tests learner's knowledge of the inverse matrix method for a 3x3 matrix.</p>"}, "preamble": {"js": "", "css": ""}, "type": "question", "contributors": [{"name": "Mark Hodds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/510/"}]}]}], "contributors": [{"name": "Mark Hodds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/510/"}]}