\\[ \\begin{eqnarray*} \\simplify[std]{A+B} &=&\\simplify[std]{{a}+{b}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{a[0][0]}+{b[0][0]}}& \\simplify[std]{{a[0][1]}+{b[0][1]}}\\\\ \\simplify[std]{{a[1][0]}+{b[1][0]}}&\\simplify[std]{{a[1][1]}+{b[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{apb}}\\\\ \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} \\simplify[std]{{p}A+{q}B} &=&\\simplify[std]{{p}{a}+{q}{b}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{p}*{a[0][0]}+{q}*{b[0][0]}}& \\simplify[std]{{p}*{a[0][1]}+{q}*{b[0][1]}}\\\\ \\simplify[std]{{p}*{a[1][0]}+{q}*{b[1][0]}}&\\simplify[std]{{p}*{a[1][1]}+{q}*{b[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{lcab}}\\\\ \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} \\simplify[std]{{p1}A+{q1}B+{r1}C} &=&\\simplify[std]{{p1}{a}+{q1}{b}+{r1}{c}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{p1}*{a[0][0]}+{q1}*{b[0][0]}+{r1}*{c[0][0]}}& \\simplify[std]{{p1}*{a[0][1]}+{q1}*{b[0][1]}+{r1}*{c[0][1]}}\\\\ \\simplify[std]{{p1}*{a[1][0]}+{q1}*{b[1][0]}+{r1}*{c[1][0]}}&\\simplify[std]{{p1}*{a[1][1]}+{q1}*{b[1][1]}+{r1}*{c[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{lcabc}}\\\\ \\end{eqnarray*} \\]
\n", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noleadingminus"]}, "parts": [{"prompt": "
$\\mathrm{A}+\\mathrm{B} = \\simplify[std]{{a}+{b}} = $ [[0]]
", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "apb", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "$\\simplify{{p}A+{q}B = {p}{a}+{q}{b}}=$ [[0]]", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "lcab", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "$\\simplify{{p1}A+{q1}B+{r1}C = {p1}{a}+{q1}{b}+{r1}{c}}=$ [[0]]
", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "lcabc", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "Let
\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}},\\;\\; C=\\simplify{{c}}\\]
Calculate the following $2 \\times 2$ matrices:
", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "matrix(repeat(repeat(random(-5..5 except 0),2),2))", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "q1": {"definition": "random(-6..6 except [0,1,-1,p1,q])", "templateType": "anything", "group": "Ungrouped variables", "name": "q1", "description": ""}, "c": {"definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0],b[0][0]]),2),2))", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0]]),2),2))", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "r1": {"definition": "random(-6..6 except [0,1,-1,p1,q1])", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}, "q": {"definition": "random(-6..6 except [0,1,-1,p])", "templateType": "anything", "group": "Ungrouped variables", "name": "q", "description": ""}, "p": {"definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "p1": {"definition": "random(2..6 except p)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}, "apb": {"definition": "a+b", "templateType": "anything", "group": "Ungrouped variables", "name": "apb", "description": ""}, "lcab": {"definition": "p*a+q*b", "templateType": "anything", "group": "Ungrouped variables", "name": "lcab", "description": ""}, "lcabc": {"definition": "p1*a+q1*b+r1*c", "templateType": "anything", "group": "Ungrouped variables", "name": "lcabc", "description": ""}}, "metadata": {"notes": "\n \t\t
8/02/2013:
\n \t\t
Finished first draft.
Linear combinations of $2 \\times 2$ matrices. Three examples.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Matrix Multiplication 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "ungrouped_variables": ["ba21", "a21", "a22", "ba22", "cb21", "b22", "b21", "cb22", "ac22", "ac21", "ab22", "ab21", "b12", "b11", "c12", "c11", "c22", "a11", "cb11", "cb12", "a12", "c21", "ba11", "ba12", "ab12", "ab11", "ac12", "ac11"], "tags": ["matrices", "matrix", "matrix multiplication", "matrix product", "multiplication of matrices", "multiplying matrices", "product of matrices"], "preamble": {"css": "", "js": ""}, "advice": "\\[ \\begin{eqnarray*} AB &=& \\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\simplify[]{{a11}{b11}+{a12}{b21}}&\\simplify[]{{a11}{b12}+{a12}{b22}}\\\\ \\simplify[]{{a21}{b11}+{a22}{b21}}&\\simplify[]{{a21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\var{ab11}&\\var{ab12}\\\\ \\var{ab21}&\\var{ab22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} BA &=& \\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\simplify[]{{b11}{a11}+{b12}{a21}}&\\simplify[]{{b11}{a12}+{b12}{a22}}\\\\ \\simplify[]{{b21}{a11}+{b22}{a21}}&\\simplify[]{{b21}{a12}+{b22}{a22}}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\var{ba11}&\\var{ba12}\\\\ \\var{ba21}&\\var{ba22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} CB &=& \\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\simplify[]{{c11}{b11}+{c12}{b21}}&\\simplify[]{{c11}{b12}+{c12}{b22}}\\\\ \\simplify[]{{c21}{b11}+{c22}{b21}}&\\simplify[]{{c21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\var{cb11}&\\var{cb12}\\\\ \\var{cb21}&\\var{cb22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} AC &=& \\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\simplify[]{{a11}{c11}+{a12}{c21}}&\\simplify[]{{a11}{c12}+{a12}{c22}}\\\\ \\simplify[]{{a21}{c11}+{a22}{c21}}&\\simplify[]{{a21}{c12}+{a22}{c22}}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\var{ac11}&\\var{ac12}\\\\ \\var{ac21}&\\var{ac22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]
", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "parts": [{"prompt": "$AB = \\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix} = $ [[0]]
", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n [ab11,ab12],\n [ab21,ab22]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "$BA = \\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}=$ [[0]]
", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n [ba11,ba12],\n [ba21,ba22]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "$CB = \\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix} \\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}=$ [[0]]
", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n [cb11,cb12],\n [cb21,cb22]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "$AC = \\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix}=$ [[0]]
", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n [ac11,ac12],\n [ac21,ac22]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n \n \nDo the following matrix problems
Let
\\[A=\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix},\\;\\;\n \n B=\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix},\\;\\;\n \n C=\\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix}\\]
Calculate the following products of these matrices:
10/07/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tDisplay of matrices looks untidy when individual components include negative numbers.
\n \t\t \t\tIs it worthwhile restricting all components of matrices to be non zero?
\n \t\t \t\tQuestion appears to be working correctly.
\n \t\t \n \t\t", "description": "Multiplication of $2 \\times 2$ matrices.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Matrix manipulation 2", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "ungrouped_variables": ["ba21", "a21", "a22", "a23", "b23", "b22", "b21", "ba22", "w3", "w2", "w1", "ba11", "a11", "p1", "q1", "q3", "q2", "ab23", "ab22", "ab21", "s3", "s2", "s1", "a12", "s7", "s6", "s5", "s4", "b12", "b13", "b11", "c13", "c12", "c11", "a33", "a32", "a31", "ba32", "ba33", "a13", "ba31", "s10", "s9", "b32", "b31", "v1", "v2", "v3", "mae", "mad", "mag", "mah", "ba12", "ba13", "b33", "p2", "p3", "ba23", "ab11", "s8", "ab31", "ab32", "ab33", "ab12", "m", "mac", "n", "q", "p", "s", "r", "u", "w", "v", "y", "x", "z", "ab13"], "tags": ["linear algebra", "mas104220122013CBA4_2", "matrix", "matrix manipulation", "matrix multiplication", "multiply matrix", "products of matrices"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "Let
\n\\[A = \\left(\\begin{array}{rrr} \\var{a11} & \\var{a12} & \\var{a13}\\\\ \\var{a21} & \\var{a22} & \\var{a23}\\\\ \\var{a31} & \\var{a32} & \\var{a33}\\\\ \\end{array}\\right),\\;\\;\\;\\; B= \\left(\\begin{array}{rrr} \\var{b11} & \\var{b12} & \\var{b13}\\\\ \\var{b21} & \\var{b22} & \\var{b23}\\\\ \\var{b31} & \\var{b32} & \\var{b33}\\\\ \\end{array}\\right),\\;\\;\\;\\; v= \\left(\\begin{array}{r} \\var{v1}\\\\ \\var{v2} \\\\ \\var{v3} \\end{array}\\right),\\;\\;\\;\\; w= \\left(\\begin{array}{r} \\var{w1}\\\\ \\var{w2} \\\\ \\var{w3} \\end{array}\\right)\\]
\nFind the following products:
\n$Av = $ [[0]]
\n$Bw = $ [[1]]
\n$BA = $ [[2]]
\n$AB = $ [[3]]
", "marks": 0, "gaps": [{"numColumns": 1, "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "3", "showCorrectAnswer": true, "correctAnswer": "matrix([[p1],[p2],[p3]])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}, {"numColumns": 1, "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "3", "showCorrectAnswer": true, "correctAnswer": "matrix([[q1],[q2],[q3]])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}, {"numColumns": "3", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "3", "showCorrectAnswer": true, "correctAnswer": "matrix([\n [ba11,ba12,ba13],\n [ba21,ba22,ba23],\n [ba31,ba32,ba33]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}, {"numColumns": "3", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "3", "showCorrectAnswer": true, "correctAnswer": "matrix([\n [ab11,ab12,ab13],\n [ab21,ab22,ab23],\n [ab31,ab32,ab33]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "Consider the following matrices together with the matrices from the first part of the question.
\n\\[\\begin{eqnarray}&C=& \\var{mac},\\;\\;\\;\\; &D=& \\var{mad},\\;\\;\\; \\;&E= &\\var{mae}\\\\&F=& \\left(\\begin{array}{rr} \\var{w1} & \\var{a12}\\\\ \\var{w2} & \\var{b23} \\\\ \\var{w3} & \\var{w2} \\\\\\var{v1} & \\var{b12}\\\\ 0 & \\var{-w2} \\end{array}\\right),\\;\\;\\;\\;&G=&\\var{mag},\\;\\;\\;\\;&H=&\\var{mah} \\end{eqnarray}\\]
\nWhich of the following products of matrices can be calculated?
\n[[0]]
\nPlease note that for every correct answer you get 0.5 marks and for every incorrect answer 0.5 is taken away. The minimum mark you can get is 0.
", "marks": 0, "gaps": [{"maxAnswers": 0, "matrix": "v", "shuffleAnswers": false, "shuffleChoices": true, "minAnswers": 0, "answers": ["Can be calculated", "Cannot be calculated"], "choices": ["$CD$
", "$DC$
", "$EF$
", "$FE$
", "$BC$
", "$AE$
", "$GH$
", "$HE$
", "$AG$
", "$GB$
"], "displayType": "radiogroup", "showCorrectAnswer": true, "minMarks": 0, "scripts": {}, "warningType": "none", "marks": 0, "type": "m_n_x", "maxMarks": 0}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\nAnswer the following questions on matrices.
\n\n ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ba21": {"definition": "b21*a11+b22*a21+b23*a31", "templateType": "anything", "group": "Ungrouped variables", "name": "ba21", "description": ""}, "a21": {"definition": "random(-1,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "a21", "description": ""}, "a22": {"definition": "random(-4,-3,-2,-1,1,2,3,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a22", "description": ""}, "a23": {"definition": "random(-2..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a23", "description": ""}, "b23": {"definition": "random(1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "b23", "description": ""}, "b22": {"definition": "random(-4,-3,-2,-1,1,2,3,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "b22", "description": ""}, "b21": {"definition": "random(0..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "b21", "description": ""}, "ba22": {"definition": "b21*a12+b22*a22+b23*a32", "templateType": "anything", "group": "Ungrouped variables", "name": "ba22", "description": ""}, "w3": {"definition": "random(-4..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "w3", "description": ""}, "w2": {"definition": "random(-4..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "w2", "description": ""}, "w1": {"definition": "random(4..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "w1", "description": ""}, "mah": {"definition": "matrix(repeat(repeat(random(-2..9),x),y))", "templateType": "anything", "group": "Ungrouped variables", "name": "mah", "description": ""}, "ba32": {"definition": "b31*a12+b32*a22+b33*a32", "templateType": "anything", "group": "Ungrouped variables", "name": "ba32", "description": ""}, "q1": {"definition": "b11*w1+b12*w2+b13*w3", "templateType": "anything", "group": "Ungrouped variables", "name": "q1", "description": ""}, "q3": {"definition": "b31*w1+b32*w2+b33*w3", "templateType": "anything", "group": "Ungrouped variables", "name": "q3", "description": ""}, "q2": {"definition": "b21*w1+b22*w2+b23*w3", "templateType": "anything", "group": "Ungrouped variables", "name": "q2", "description": ""}, "ab23": {"definition": "a21*b13+a22*b23+a23*b33", "templateType": "anything", "group": "Ungrouped variables", "name": "ab23", "description": ""}, "ab22": {"definition": "a21*b12+a22*b22+a23*b32", "templateType": "anything", "group": "Ungrouped variables", "name": "ab22", "description": ""}, "ab21": {"definition": "a21*b11+a22*b21+a23*b31", "templateType": "anything", "group": "Ungrouped variables", "name": "ab21", "description": ""}, "s3": {"definition": "if(s=5,0.5,-0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "if(q=m,0.5,-0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "if(n=p,0.5,-0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "ba31": {"definition": "b31*a11+b32*a21+b33*a31", "templateType": "anything", "group": "Ungrouped variables", "name": "ba31", "description": ""}, "s7": {"definition": "if(u=y,0.5,-0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "s7", "description": ""}, "s6": {"definition": "if(r=3,0.5,-0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "s6", "description": ""}, "s5": {"definition": "if(m=3,0.5,-0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "if(r=2,0.5,-0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}, "b12": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped 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5/07/2012:
\n \t\t \t\t \t\tAdded tags.
\n \t\t \t\t \t\tQuestion appears to be working correctly.
\n \t\t \t\t \t\t\n \t\t \t\t \n \t\t \n \t\t", "description": "
Exercises in multiplying matrices.
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