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{"name": "Maria's copy of Find the Determinant of a 4 x 4 Matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Maria's copy of Find the Determinant of a 4 x 4 Matrix", "tags": [], "metadata": {"description": "<p>Find the determinant of a&nbsp;$4 \\times 4$ matrix.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Consider the&nbsp;$4 \\times 4$ matrix,</p>\n<p>\\begin{align} \\mathrm{A} &amp;= \\var{a} \\end{align}</p>", "advice": "<p>The determinant of an $n \\times n$ matrix an be calculated in terms of the determinant of minor matrices of size $[n-1] \\times [n-1]$.&nbsp;</p>\n<p>For our example we multiply&nbsp;each value in a chosen row of the matrix by its cofactor,</p>\n<p>\\[&nbsp;(-1)^{i+j} M_{ij} \\]</p>\n<p>for a&nbsp;value at row $i$ and column $j$, where $M_{ij}$ is its&nbsp;minor $3 \\times 3$ matrix. Adding these values together gives the determinant of our $4 \\times 4$ matrix.</p>\n<p>In our example, choosing the first row we have,</p>\n<p>\\[&nbsp;\\det\\left(\\mathrm{A}\\right) = \\var{a11} \\cdot \\det \\var{m1}-\\var{a12} \\cdot \\det \\var{m2}+\\var{a13} \\cdot \\det \\var{m3} - \\var{a14} \\cdot \\det \\var{m4} \\]</p>\n<p>Notice that we do not need to compute all of the 3 x 3 matrices, particularly if the starting row is chosen carefully.</p>\n<p>Then,</p>\n<p>\\[&nbsp;\\det\\left(\\mathrm{A}\\right) = (\\simplify[]{{a11}*{det(m1)}})+(\\simplify[]{{a14}*{det(m4)}}) = \\var{determinant} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "extensions": [], "variables": {"a24": {"name": "a24", "group": "Matrix Elements", "definition": "0", "description": "", "templateType": "anything"}, "a33": {"name": "a33", "group": "Matrix Elements", "definition": "random(-6..6 except 0)", "description": "<p>Matrix element</p>", "templateType": "anything"}, "m4": {"name": "m4", "group": "Minor Matrices", "definition": "matrix([ [a21,a22,a23],[a31,a32,a33],[a41,a42,a43] ])", "description": "<p>submatrix</p>", "templateType": "anything"}, "a42": {"name": "a42", "group": "Matrix Elements", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "a31": {"name": "a31", "group": "Matrix Elements", "definition": "0", "description": "", "templateType": "anything"}, "a43": {"name": "a43", "group": "Matrix Elements", "definition": "0", "description": "", "templateType": "anything"}, "a23": {"name": "a23", "group": "Matrix Elements", "definition": "random(-5..5)", "description": "<p>Matrix element</p>", "templateType": "anything"}, "a": {"name": "a", "group": "Given Matrix", "definition": "matrix([ [a11,a12,a13,a14],[a21,a22,a23,a24],[a31,a32,a33,a34],[a41,a42,a43,a44] ])", "description": "<p>4 x 4 matrix provided to student</p>", "templateType": "anything"}, "a44": {"name": "a44", "group": "Matrix Elements", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything"}, "a12": {"name": "a12", "group": "Matrix Elements", "definition": "0", "description": "", "templateType": "anything"}, "m2": {"name": "m2", "group": "Minor Matrices", "definition": "matrix([ [a21,a23,a24],[a31,a33,a34],[a41,a43,a44] ])", "description": "<p>Submatrix</p>", "templateType": "anything"}, "determinant": {"name": "determinant", "group": "Ungrouped variables", "definition": "a11*det(m1)+a14*det(m4)", "description": "<p>Determinant of the given matrix</p>", "templateType": "anything"}, "a41": {"name": "a41", "group": "Matrix Elements", "definition": "0", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "Matrix Elements", "definition": "random(-6..6 except 0)", "description": "<p>Matrix element</p>", "templateType": "anything"}, "m1": {"name": "m1", "group": "Minor Matrices", "definition": "matrix([ [a22,a23,a24],[a32,a33,a34],[a42,a43,a44] ])", "description": "<p>Submatrix</p>", "templateType": "anything"}, "m3": {"name": "m3", "group": "Minor Matrices", "definition": "matrix([ [a21,a22,a24],[a31,a32,a34],[a41,a42,a44] ])", "description": "<p>Submatrix</p>", "templateType": "anything"}, "aa": {"name": "aa", "group": "Ungrouped variables", "definition": "0", "description": "<p>Matrix element</p>", "templateType": "anything"}, "a11": {"name": "a11", "group": "Matrix Elements", "definition": "random(-6..6 except 0)", "description": "<p>Matrix element</p>", "templateType": "anything"}, "a13": {"name": "a13", "group": "Matrix Elements", "definition": "0", "description": "", "templateType": "anything"}, "a14": {"name": "a14", "group": "Matrix Elements", "definition": "random(-6..6 except 0) ", "description": "", "templateType": "anything"}, "a32": {"name": "a32", "group": "Matrix Elements", "definition": "0", "description": "", "templateType": "anything"}, "a34": {"name": "a34", "group": "Matrix Elements", "definition": "random(-5..5)", "description": "<p>Matrix element</p>", "templateType": "anything"}, "a21": {"name": "a21", "group": "Matrix Elements", "definition": "0", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["aa", "determinant"], "variable_groups": [{"name": "Given Matrix", "variables": ["a"]}, {"name": "Minor Matrices", "variables": ["m1", "m2", "m3", "m4"]}, {"name": "Matrix Elements", "variables": ["a11", "a12", "a13", "a14", "a21", "a22", "a23", "a24", "a31", "a32", "a33", "a34", "a41", "a42", "a43", "a44"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Calculate the determinant of the matrix through expansion by&nbsp;the first row.</p>\n<p>$\\det&nbsp;\\left(\\mathrm{A} \\right) = $&nbsp;[[0]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2.0", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "determinant", "maxValue": "determinant", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}