// Numbas version: finer_feedback_settings {"name": "Clodagh's copy of Properties of the transpose", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "

If $A = \\begin{pmatrix} 1 & 0 & -1\\\\ 2 & 3 & 4  \\end{pmatrix}$ and $B = \\begin{pmatrix} 1 & 2\\\\ 2 & 1\\\\ 3 & 2  \\end{pmatrix}$, verify that $(AB)^T = B^T A^T$.

", "tags": [], "name": "Clodagh's copy of Properties of the transpose", "parts": [{"sortAnswers": false, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "prompt": "

Increase/decrease the number of rows or columns if necessary:

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$(AB)^T = $ [[0]]

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$B^TA^T = $ [[1]]

", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "marks": 0, "scripts": {}, "customName": "", "gaps": [{"adaptiveMarkingPenalty": 0, "variableReplacements": [], "numRows": "2", "showFeedbackIcon": true, "customMarkingAlgorithm": "malrules:\n [\n [\"matrix([-2,0],[20,15])\",\"Did you remember to transpose your answer?\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\n\ncorrect_cells_malrules (The indexes of the cells which are correct):\n filter(\n if(p[0]With the aid of this result (in other words, using the fact that $(AB)^T = B^TA^T$ for any matrices $A$ and $B$ where $AB$ can be calculated) simplify the expressions below, where $X$ and $Y$ are arbitrary (i.e. unspecified) matrices:

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Note: Whenever you wish Numbas to multiply two variables, you need to use the multiplication sign, so for $AB$, write $A*B$ etc

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$(XY^T)^T = $ [[0]]

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$(X^TY^T)^T = $ [[1]]

", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "marks": 0, "scripts": {}, "customName": "", "gaps": [{"checkingType": "absdiff", "adaptiveMarkingPenalty": 0, "variableReplacements": [], "answer": "Y*X^T", "showFeedbackIcon": true, "customMarkingAlgorithm": "malrules:\n [\n [\"X^T*Y\",\"Be very careful with the order of multiplication. Remember when you apply the transpose to a bracketed expression, you must reverse the order\u00a0in which the matrices are being multiplied.\"], \n [\"X^T*Y^T\",\"Be very careful with the order of multiplication. Remember when you apply the transpose to a bracketed expression, you must reverse the order\u00a0in which the matrices are being multiplied. Also, note that there was already a $Y^T$ inside the bracket. What is $(Y^T)^T$?\"],\n [\"Y^T*X^T\",\"Almost there - you remembered to reverse the order of multiplication when applying the transpose to the bracketed expression - well done. However, note that there was already a $Y^T$ inside the bracket. What is $(Y^T)^T$?\"],\n [\"X*Y^T\",\"You haven't applied the transpose that was outside the bracket.\"],\n [\"Y*X\",\"Did you remember to apply the transpose operator to both matrices?\"],\n [\"X*Y\",\"Also, did you remember to apply the transpose operator to both matrices?\"]\n ]\n\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))