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{"name": "Matrices: Multiplication 03", "extensions": [], "custom_part_types": [], "resources": [["question-resources/matrixMult.png", "/srv/numbas/media/question-resources/matrixMult.png"], ["question-resources/Matrix_Multiplication_02.gif", "/srv/numbas/media/question-resources/Matrix_Multiplication_02.gif"], ["question-resources/Matrix_Multiplication_01.gif", "/srv/numbas/media/question-resources/Matrix_Multiplication_01.gif"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Matrices: Multiplication 03", "tags": [], "metadata": {"description": "<p>Multiplying matrices (pre-defined sizes in answers)</p>\n<p>This set is designed to emphasise non-commutativity.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>An $n \\times p$ matrix $A$ can be multiplied by a $p \\times n$ matrix $B$ to form an $n \\times m$ matrix $AB=C$.</p>\n<p>The number of columns of $A$ must match the number of rows of $B$.</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/matrixMult.png\" width=\"308\" height=\"70\"/></p>\n<p>The element in the $i^{th}$ row and $j^{th}$ column of $C$ is obtained&nbsp;by multiplying the $i^{th}$ row of $A$ with the $j^{th}$ column of $B$.</p>", "advice": "<h4>We are asked to carry out matix multiplications.</h4>\n<p></p>\n<p>First of all, you should always check that the multiplication is even possible. Write down the dimensions (in order) of the two matrices:</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/Matrix_Multiplication_01.gif\" width=\"275\" height=\"65\"/></p>\n<p>The number of columns in the first must match the number of rows in the second. As a bonus this will also give you the dimensions of the product matrix.</p>\n<p>The multiplication is then carried out moving across the rows of the first matrix and down the columns of the second:</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/Matrix_Multiplication_02.gif\" width=\"308\" height=\"128\"/></p>\n<p style=\"text-align: center;\"></p>\n<p>Using this techniques will give:</p>\n<p>$A_1A_2=\\var{A1}\\var{A2}$</p>\n<p></p>\n<p style=\"text-align: center;\">$A_1 A_2=\\var{prodA}$&nbsp;</p>\n<p></p>\n<p>$A_2 A_1=\\var{A2}\\var{A1}$</p>\n<p style=\"text-align: center;\">$A_2 A_1=\\var{prodA2}$&nbsp;</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>$B_1B_2=\\var{B1}\\var{B2}$</p>\n<p style=\"text-align: center;\">$B_1B_2=\\var{prodB}$</p>\n<p>&nbsp;</p>\n<p>$B_2B_1=\\var{B2}\\var{B1}$</p>\n<p style=\"text-align: center;\">$B_2B_1=\\var{prodB2}$</p>\n<p>&nbsp;</p>\n<p>Hopefully, you can see that when the multiplication is reversed we get a different answer! That is very different to what we see in conventional arithmatic.</p>\n<p>With \"normal\" multiplication&nbsp; &nbsp;$3 \\times 2 = 2 \\times 3$, we say that multiplication is commutative - the order does not matter. That is not the case with matrix multiplication.</p>\n<p>It is clear that $AB$ and $BA$ are <strong>not</strong> in general the same. In fact it is the <strong>exception</strong> that $AB = BA$.&nbsp;In the special case in which $AB = BA$ we say that the matrices $A$ and $B$ <strong>commute</strong>.</p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"n1": {"name": "n1", "group": "Matrix A", "definition": "2", "description": "", "templateType": "number"}, "m1": {"name": "m1", "group": "Matrix A", "definition": "2", "description": "", "templateType": "number"}, "A1": {"name": "A1", "group": "Matrix A", "definition": "transpose(matrix(repeat(repeat(random(0..9),n1),m1)))", "description": "", "templateType": "anything"}, "A2": {"name": "A2", "group": "Matrix A", "definition": "transpose(matrix(repeat(repeat(random(0..9),m1),n1)))", "description": "", "templateType": "anything"}, "ProdA": {"name": "ProdA", "group": "Matrix A", "definition": "{A1}{A2}", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "Matrix B", "definition": "3", "description": "", "templateType": "number"}, "m2": {"name": "m2", "group": "Matrix B", "definition": "3", "description": "", "templateType": "number"}, "B1": {"name": "B1", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),n2),m2)))", "description": "", "templateType": "anything"}, "B2": {"name": "B2", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),m2),n2)))", "description": "", "templateType": "anything"}, "ProdB": {"name": "ProdB", "group": "Matrix B", "definition": "{B1}{B2}", "description": "", "templateType": "anything"}, "n3": {"name": "n3", "group": "Matrix C", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "m3": {"name": "m3", "group": "Matrix C", "definition": "random(1..1)", "description": "", "templateType": "anything"}, "C1": {"name": "C1", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(0..9),n3),m3)))", "description": "", "templateType": "anything"}, "m3a": {"name": "m3a", "group": "Matrix C", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "C2": {"name": "C2", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(0..9),m3),m3a)))", "description": "", "templateType": "anything"}, "prodC": {"name": "prodC", "group": "Matrix C", "definition": "{C1}{C2}", "description": "", "templateType": "anything"}, "ProdA2": {"name": "ProdA2", "group": "Matrix A", "definition": "{A2}{A1}", "description": "", "templateType": "anything"}, "ProdB2": {"name": "ProdB2", "group": "Matrix B", "definition": "{B2}{B1}", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Matrix A", "variables": ["n1", "m1", "A1", "A2", "ProdA", "ProdA2"]}, {"name": "Matrix B", "variables": ["n2", "m2", "B1", "B2", "ProdB", "ProdB2"]}, {"name": "Matrix C", "variables": ["n3", "m3", "C1", "m3a", "C2", "prodC"]}], "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": "<h4>Carry out the following multiplications:</h4>\n<p>&nbsp;</p>\n<p>$A_1A_2=\\var{A1}\\var{A2}$</p>\n<p style=\"text-align: center;\">$A_1 A_2=$ [[0]]</p>\n<p></p>\n<p>$A_2 A_1=\\var{A2}\\var{A1}$</p>\n<p style=\"text-align: center;\">$A_2 A_1=$ [[1]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>$B_1B_2=\\var{B1}\\var{B2}$</p>\n<p style=\"text-align: center;\">$B_1B_2=$ [[2]]</p>\n<p style=\"text-align: left;\">&nbsp;&nbsp;</p>\n<p>$B_2B_1=\\var{B2}\\var{B1}$</p>\n<p style=\"text-align: center;\">$B_2B_1=$ [[3]]</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">From these results, what can you conclude about matrix multiplication in general?</p>\n<p style=\"text-align: center;\">[[4]]</p>\n<p style=\"text-align: left;\"></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": "{ProdA}", "correctAnswerFractions": false, "numRows": "{n1}", "numColumns": "{n1}", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", 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