// Numbas version: finer_feedback_settings
{"name": "Matrix arithmetic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Matrix arithmetic", "statement": "<p>Given the matrix:</p>\n<p style=\"padding-left: 30px;\">\\(A=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "extensions": [], "metadata": {"description": "<p>This question tests students knowledge of basic matrix arithmetic.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "preamble": {"css": "", "js": ""}, "advice": "<p style=\"padding-left: 30px;\">\\(A=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(A^2=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">Remember multiplication of matrices is carried out by multiplying the rows of the first matrix by the columns of the second matrix.</p>\n<p style=\"padding-left: 30px;\">\\(A^2=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}=\\begin{pmatrix}\\var{a11}*\\var{a11}+\\var{a12}*\\var{a21}&amp;\\var{a11}*\\var{a12}+\\var{a12}*\\var{a22}\\\\ \\var{a21}*\\var{a11}+\\var{a22}*\\var{a21}&amp;\\var{a21}*\\var{a12}+\\var{a22}*\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(A^2=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&amp;\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{k1}A=\\begin{pmatrix} \\var{k1}*\\var{a11}&amp; \\var{k1}*\\var{a12}\\\\ \\var{k1}*\\var{a21}&amp;\\var{k1}*\\var{a22}\\end{pmatrix}=\\begin{pmatrix} \\simplify{{k1}*{a11}}&amp; \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&amp;\\simplify{{k1}*{a22}}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\left(\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&amp;\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\simplify{{k1}*{a11}}&amp; \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&amp;\\simplify{{k1}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\var{k2}&amp;0\\\\0&amp;\\var{k2}\\end{pmatrix}\\right)^t\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&amp;\\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}^t\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&amp;\\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}\\\\ \\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}\\)</p>", "variable_groups": [], "variables": {"b22": {"name": "b22", "definition": "{a21}*{a12}+{a22}^2+{k1}*{a22}+{k2}", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "b11": {"name": "b11", "definition": "{a11}^2+{a12}*{a21}+{k1}*{a11}+{k2}", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "a12": {"name": "a12", "definition": "random(0..10#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "k2": {"name": "k2", "definition": "random(6..12#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "b21": {"name": "b21", "definition": "{a12}*{a11}+{a22}*{a12}+{k1}*{a12}", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "a22": {"name": "a22", "definition": "random(11..21#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "a11": {"name": "a11", "definition": "random(1..10#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "a21": {"name": "a21", "definition": "random(2..9#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "k1": {"name": "k1", "definition": "random(2..7#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "b12": {"name": "b12", "definition": "{a21}*{a11}+{a22}*{a21}+{k1}*{a21}", "description": "", "templateType": "anything", "group": "Ungrouped variables"}}, "tags": [], "functions": {}, "ungrouped_variables": ["a11", "a12", "a21", "a22", "k1", "k2", "b11", "b12", "b21", "b22"], "parts": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "marks": 0, "gaps": [{"allowFractions": false, "variableReplacements": [], "allowResize": false, "correctAnswerFractions": false, "showCorrectAnswer": true, "marks": "2", "numRows": "2", "numColumns": "2", "showFeedbackIcon": true, "markPerCell": false, "type": "matrix", "tolerance": 0, "correctAnswer": "matrix([\n   [b11,b12],\n   [b21,b22]\n])    ", "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "scripts": {}, "prompt": "<p>Evaluate the following expression:</p>\n<p style=\"padding-left: 60px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t\\) =&nbsp;[[0]]</p>", "showCorrectAnswer": true}], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}