// Numbas version: exam_results_page_options
{"name": "Minimalpolynom und Potenzen einer Matrix", "extensions": ["linearalgebra2", "polynomials"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Minimalpolynom und Potenzen einer Matrix", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Gegeben ist die Matrix</p>\n<p>\\[ A = \\var[fractionNumbers]{A}\\in M_{\\var{size}}(\\mathbb R). \\]</p>\n<p>Es gilt ${\\rm minpol}_A = \\var{minpol}$.</p>", "advice": "<p>Es gilt ${\\rm minpol}_A(A) =0$, das bedeutet $A^{\\var{degree(minpol)}} = \\var{-c} E_{\\var{size}}$, also $A^{\\var{r}} = \\var{-c}A$.</p>", "rulesets": {}, "extensions": ["linearalgebra2", "polynomials"], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "S*companion_matrix(map(minpol[x], x, 0..degree(minpol)-1), 3)*inverse_matrix(S)", "description": "", "templateType": "anything"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "degree(minpol)+1", "description": "", "templateType": "anything"}, "size": {"name": "size", "group": "Ungrouped variables", "definition": "3", "description": "", "templateType": "anything"}, "minpol": {"name": "minpol", "group": "Ungrouped variables", "definition": "polynomial(x, [c] + repeat(0, size-1) +[1])", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "templateType": "anything"}, "Ar": {"name": "Ar", "group": "Ungrouped variables", "definition": "matrix_power(A, r)", "description": "", "templateType": "anything"}, "S": {"name": "S", "group": "Ungrouped variables", "definition": "random_trafo(3, [1,2,3,3,2,1,2,1,1,2,1,3])", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "r", "size", "minpol", "c", "Ar", "S"], "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>Berechnen Sie:</p>\n<p>$A^{\\var{r}} = $ [[0]].</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>Sie m&uuml;ssen keine einzige Matrixmultiplikation ausf&uuml;hren, um das Ergebnis zu finden!</p>"}], "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "{Ar}", "correctAnswerFractions": true, "numRows": "size", "numColumns": "size", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}]}], "contributors": [{"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}