$AB = \\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\end{pmatrix} = $ [[0]]
", "gaps": [{"tolerance": 0, "numRows": "2", "scripts": {}, "allowResize": false, "allowFractions": false, "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "markPerCell": false, "showCorrectAnswer": true, "marks": "1", "correctAnswer": "matrix([\n [ab11,ab12],\n [ab21,ab22]\n])", "showFeedbackIcon": true, "numColumns": "2", "type": "matrix", "variableReplacements": []}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true}, {"marks": 0, "type": "gapfill", "scripts": {}, "showFeedbackIcon": true, "prompt": "$BA = \\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\end{pmatrix}\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23}\\\\ \\end{pmatrix}=$ [[0]]
", "gaps": [{"tolerance": 0, "numRows": "3", "scripts": {}, "allowResize": false, "allowFractions": false, "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "markPerCell": false, "showCorrectAnswer": true, "marks": "1", "correctAnswer": "matrix([\n [ba11,ba12,ba13],\n [ba21,ba22,ba23],\n [ba31,ba32,ba33]\n])", "showFeedbackIcon": true, "numColumns": "3", "type": "matrix", "variableReplacements": []}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true}], "statement": "Do the following matrix problems :
Let
\\[A=\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23}\\\\ \\end{pmatrix},\\;\\; B=\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\end{pmatrix},\\;\\; \\]
Calculate the following products of these matrices:
\\[ \\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*} \\]
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"description": ""}, "a11": {"definition": "random(-2,1,2)", "name": "a11", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a21": {"definition": "random(-2..2)", "name": "a21", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ba21": {"definition": "b21*a11+b22*a21", "name": "ba21", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ac12": {"definition": "a11*c12+a12*c22", "name": "ac12", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ba12": {"definition": "b11*a12+b12*a22", "name": "ba12", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a13": {"definition": "random(0..2)", "name": "a13", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c12": {"definition": "a12+b12", "name": "c12", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ab12": {"definition": "a11*b12+a12*b22+a13*b32", "name": "ab12", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ac22": {"definition": "a21*c12+a22*c22", "name": "ac22", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b32": {"definition": "random(2..5)", "name": "b32", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a22": {"definition": "random(1..3)", "name": "a22", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c21": {"definition": "random(2..5)", "name": "c21", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "cb12": {"definition": "c11*b12+c12*b22", "name": "cb12", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ba13": {"definition": "b11*a13+b12*a23", "name": "ba13", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ab11": {"definition": "a11*b11+a12*b21+a13*b31", "name": "ab11", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a12": {"definition": "random(1..4)", "name": "a12", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "cb22": {"definition": "c21*b12+c22*b22", "name": "cb22", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b22": {"definition": "random(-3..-1)", "name": "b22", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ba23": {"definition": "b21*a13+b22*a23", "name": "ba23", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b31": {"definition": "random(-1..1)", "name": "b31", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b21": {"definition": "random(2,3)", "name": "b21", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b11": {"definition": "random(-3,-1,0,3)", "name": "b11", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ba11": {"definition": "b11*a11+b12*a21", "name": "ba11", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c22": {"definition": "random(0,1)", "name": "c22", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ba22": {"definition": "b21*a12+b22*a22", "name": "ba22", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ab21": {"definition": "a21*b11+a22*b21+a23*b31", "name": "ab21", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ba31": {"definition": "b31*a11+b32*a21", "name": "ba31", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c11": {"definition": "random(1,0,4)", "name": "c11", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ba33": {"definition": "b31*a13+b32*a23", "name": "ba33", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b12": {"definition": "random(-3..1)", "name": "b12", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "name": "Matrices: Multiplication", "tags": [], "extensions": [], "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "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", "a13", "a23", "b31", "b32", "ba13", "ba23", "ba31", "ba32", "ba33"], "functions": {}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Multiplication of $2 \\times 2$ matrices.
"}, "type": "question", "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}]}]}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}]}