// Numbas version: exam_results_page_options
{"name": "Matrizenrechnung", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Matrizenrechnung", "tags": [], "metadata": {"description": "<p>Zwei einfache Aufgaben zur Matrizenrechnung.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h3>Matrizenrechnung</h3>\n<p>Berechnen Sie die unten angegebenen Matrizen.</p>\n<p><em>Achten Sie bitte darauf, ggf. zun&auml;chst die korrekte Gr&ouml;&szlig;e der Ergebnis-Matrix festzulegen!</em></p>", "advice": "<p><strong>L&ouml;sung zu a)</strong></p>\n<p>Bei der transponierten Matrix werden einfach nur Zeilen und Spalten vertauscht, das Ergebnis ist demnach</p>\n<p>$\\var{B}^T=\\var{BT}$.</p>\n<p><strong>L&ouml;sung zu b):</strong></p>\n<p>Allgemein gilt: Ist $A$ eine $(n,p)$-Matrix und $B$ eine $(p,m)$-Matrix dann ist die Produktmatrix $C = A \\cdot&nbsp; B$ ist eine&nbsp;$(m,n)$-Matrix, und die Elemente $c_{ik}$ ergeben sich als <em>Skalarprodukt</em> der&nbsp;$i$-ten Zeile von $A$ mit der $k$-ten Spalte von $B$, also:<br/> $c_{ik} \\ = \\ \\sum\\limits_{r=1}^p a_{ir} b_{rk} \\qquad(i=1,\\ldots,m; \\ k=1,\\ldots,n).$</p>\n<p>Im konkreten Fall ergibt dies:&nbsp;</p>\n<p>$A$ ist eine $(3,3)$-Matrix und $B$ ist eine $(3,2)$-Matrix, also ist $C=A\\cdot B$ eine $(3,2)$-Matrix, es ergibt sich dann:</p>\n<p>$\\var{A}\\cdot\\var{B}=\\begin{pmatrix}<br/>\\var{A[0][0]}\\cdot \\var{B[0][0]}+&nbsp;\\var{A[0][1]}\\cdot \\var{B[1][0]} +&nbsp; \\var{A[0][2]}\\cdot \\var{B[2][0]} &amp;&nbsp;<br/>\\var{A[0][0]}\\cdot \\var{B[0][1]}+&nbsp;\\var{A[0][1]}\\cdot \\var{B[1][1]} +&nbsp; \\var{A[0][2]}\\cdot \\var{B[2][1]}&nbsp; \\\\<br/>\\var{A[1][0]}\\cdot \\var{B[0][0]}+&nbsp;\\var{A[1][1]}\\cdot \\var{B[1][0]} +&nbsp; \\var{A[1][2]}\\cdot \\var{B[2][0]} &amp;&nbsp;<br/>\\var{A[1][0]}\\cdot \\var{B[0][1]}+&nbsp;\\var{A[1][1]}\\cdot \\var{B[1][1]} +&nbsp; \\var{A[1][2]}\\cdot \\var{B[2][1]}&nbsp; \\\\<br/>\\var{A[2][0]}\\cdot \\var{B[0][0]}+&nbsp;\\var{A[2][1]}\\cdot \\var{B[1][0]} +&nbsp; \\var{A[2][2]}\\cdot \\var{B[2][0]} &amp;&nbsp;<br/>\\var{A[2][0]}\\cdot \\var{B[0][1]}+&nbsp;\\var{A[2][1]}\\cdot \\var{B[1][1]} +&nbsp; \\var{A[2][2]}\\cdot \\var{B[2][1]}&nbsp; \\\\<br/>\\end{pmatrix}=\\var{AxB}$</p>", "rulesets": {}, "extensions": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "matrix(\n  repeat(repeat(random(0..9),3), 3)\n)", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "matrix(\n  repeat(repeat(random(0..9),2), 3)\n)", "description": "", "templateType": "anything"}, "AxB": {"name": "AxB", "group": "Ungrouped variables", "definition": "A*B", "description": "", "templateType": "anything"}, "BT": {"name": "BT", "group": "Ungrouped variables", "definition": "transpose(B)", "description": "", "templateType": "anything"}, "square": {"name": "square", "group": "Ungrouped variables", "definition": "[[2,2],[1,1]]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "B", "AxB", "BT", "square"], "variable_groups": [], "functions": {}, "preamble": {"js": "\n\n", "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>$\\var{B}^T=$ [[0]]</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": "BT", "correctAnswerFractions": false, "numRows": "2", "numColumns": "3", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false}], "sortAnswers": false}, {"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, "prompt": "<p>$\\var{A}\\cdot\\var{B}=$</p>", "correctAnswer": "AxB", "correctAnswerFractions": false, "numRows": "1", "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}]}], "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}