// Numbas version: finer_feedback_settings
{"name": "MATH6058 Matrix Multiplication 1 (2x2 matrices)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "MATH6058 Matrix Multiplication 1 (2x2 matrices)", "tags": [], "metadata": {"description": "<p>Multiplication of $2 \\times 2$ matrices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Do the following matrix problems <br/>Let <br/>\\[A=\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix},\\;\\; B=\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix},\\;\\; C=\\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}\\]<br/>Calculate the following products of these matrices:</p>", "advice": "<h4>a)</h4>\n<p>\\[ \\begin{eqnarray*} AB &amp;=&amp; \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{a11}{b11}+{a12}{b21}}&amp;\\simplify[]{{a11}{b12}+{a12}{b22}}\\\\ \\simplify[]{{a21}{b11}+{a22}{b21}}&amp;\\simplify[]{{a21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ab11}&amp;\\var{ab12}\\\\ \\var{ab21}&amp;\\var{ab22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>b)</h4>\n<p>\\[ \\begin{eqnarray*} BA &amp;=&amp; \\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{b11}{a11}+{b12}{a21}}&amp;\\simplify[]{{b11}{a12}+{b12}{a22}}\\\\ \\simplify[]{{b21}{a11}+{b22}{a21}}&amp;\\simplify[]{{b21}{a12}+{b22}{a22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ba11}&amp;\\var{ba12}\\\\ \\var{ba21}&amp;\\var{ba22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>c)</h4>\n<p>\\[ \\begin{eqnarray*} CB &amp;=&amp; \\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{c11}{b11}+{c12}{b21}}&amp;\\simplify[]{{c11}{b12}+{c12}{b22}}\\\\ \\simplify[]{{c21}{b11}+{c22}{b21}}&amp;\\simplify[]{{c21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{cb11}&amp;\\var{cb12}\\\\ \\var{cb21}&amp;\\var{cb22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>d)</h4>\n<p>\\[ \\begin{eqnarray*} AC &amp;=&amp; \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{a11}{c11}+{a12}{c21}}&amp;\\simplify[]{{a11}{c12}+{a12}{c22}}\\\\ \\simplify[]{{a21}{c11}+{a22}{c21}}&amp;\\simplify[]{{a21}{c12}+{a22}{c22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ac11}&amp;\\var{ac12}\\\\ \\var{ac21}&amp;\\var{ac22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ba11": {"name": "ba11", "group": "Ungrouped variables", "definition": "b11*a11+b12*a21", "description": "", "templateType": "anything", "can_override": false}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "ba12": {"name": "ba12", "group": "Ungrouped variables", "definition": "b11*a12+b12*a22", "description": "", "templateType": "anything", "can_override": false}, "ba21": {"name": "ba21", 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\\end{pmatrix}\\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}=$&nbsp;[[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": "matrix([\n    [ac11,ac12],\n    [ac21,ac22]\n])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}]}], "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}